100 + 100 ≠ 200
First off -- you do not need a 200A service to feed two 100A panelboards, provided the total load as determined by NEC Article 220's calculations does not exceed the service ampacity. This is a consequence of the how 230.90(A) Exception 3 interacts with 230.40 Exception 2 and your specific setup (bold running text mine for emphasis, italics theirs):
230.90 Where Required.
Each ungrounded service conductor shall have overload protection.
(A) Ungrounded Conductor. Such protection shall be
provided by an overcurrent device in series with each ungrounded service conductor that has a rating or setting not
higher than the allowable ampacity of the conductor. A set
of fuses shall be considered all the fuses required to protect
all the ungrounded conductors of a circuit. Single-pole circuit breakers, grouped in accordance with 230.71(B), shall
be considered as one protective device.
Exception No. 1: For motor-starting currents, ratings that
comply with 430.52, 430.62, and 430.63 shall be permitted.
Exception No.2: Fuses and circuit breakers with a rating
or setting that complies with 240.4(B) or (C) and 240.6
shall be permitted.
Exception No.3: Two to six circuit breakers or sets of
fuses shall be permitted as the overcurrent device to provide the overload protection. The sum of the ratings of the
circuit breakers or fuses shall be permitted to exceed the
ampacity of the service conductors, provided the calculated
load does not exceed the ampacity of the service
conductors.
Exception No.4: Overload protection for fire pump supply
conductors shall comply with 695.4(B)(2)(a).
Exception No.5: Overload protection for 120/240-volt,
3-wire, single-phase dwelling services shall be permitted in
accordance with the requirements of 310.15(B)(7).
(Note that exception 5 has to do with the 310.15(B)(7) allowances for residential service conductor sizing -- they need to be taken into account when determining the total ampacity of your service and the ampacity of your individual sets of service entrance conductors, but stop there.)
A tale of a misplaced panelboard
But, there's more! While what the electrician did (putting a second main panel on your service) would have been OK when done properly as it would fall under 2014 NEC 230.40, exception 2 (bold running text mine for emphasis, italics theirs):
230.40 Number of Service-Entrance Conductor Sets.
Each service drop, set of overhead service conductors, set
of underground service conductors, or service lateral shall
supply only one set of service-entrance conductors.
Exception No.1: A building with more than one occupancy shall be permitted to have one set of service-
entrance conductors for each service, as defined in 230.2,
run to each occupancy or group of occupancies. If the
number of service disconnect locations for any given classification of service does not exceed six, the requirements of
230.2(E) shall apply at each location. If the number of
service disconnect locations exceeds six for any given supply classification, all service disconnect locations for all
supply characteristics, together with any branch circuit or
feeder supply sources, if applicable, shall be clearly described using suitable graphics or text, or both, on one or
more plaques located in an approved, readily accessible
location(s) on the building or structure served and as near
as practicable to the point(s) of attachment or entry(ies) for
each service drop or service lateral, and for each set of
overhead or underground service conductors.
Exception No.2: Where two to six service disconnecting
means in separate enclosures are grouped at one location
and supply separate loads from one service drop, set of
overhead service conductors, set of underground service
conductors, or service lateral, one set of service-entrance
conductors shall be permitted to supply each or several
such service equipment enclosures.
Exception No.3: A single-family dwelling unit and its
accessory structures shall be permitted to have one set of
service-entrance conductors run to each from a single service drop, set of overhead service conductors, set of underground service conductors, or service lateral.
Exception No.4: Two-family dwellings, multi-family dwellings, and multiple occupancy buildings shall be permitted
to have one set of service-entrance conductors installed to
supply the circuits covered in 210.25.
Exception No.5: One set of service-entrance conductors
connected to the supply side of the normal service disconnecting means shall be permitted to supply each or several
systems covered by 230.82(5) or 230.82(6).
However, your electrician screwed up when he put the second panel on the outside of your garage, as that ruins the grouping required by 230.40, exception 2 and 230.72(A):
230.72 Grouping of Disconnects.
(A) General. The two to six disconnects as permitted in
230.71 shall be grouped. Each disconnect shall be marked
to indicate the load served.
So, in any case, you'll need to have the electrician move the second panelboard inside to a spot next to where the first one lives, or replace the first panelboard with an exterior unit that can be mounted next to where the new one lives.
Good news, your article 220 load calculations are correct
Upon reviewing 220.82 and your panel photo -- your article 220 load calculation seems to be spot-on in terms of what it includes. Note that you do not need to use the 125% continuous load multiplier for Article 220 purposes as the applicable Code text explicitly refers to the nameplate rating of the appliance.
As to the tapping...
Your existing feeder conductor is indeed correctly sized for 100A -- the 83% rules for dwelling unit feeders/services in 310.15(B)(7) means that you can get by with a cable that can carry 83A, and 2AWG aluminum SE is allowed to carry 90A when terminated on 75°C lugs as per 338.10(B)(4)(a) (text from 2017 NEC, 334.80 is the section that governs temperature ratings and allowable ampacity of NM cables):
(4) Installation Methods for Branch Circuits and Feeders.
(a) Interior Installations. In addition to the provisions of
this article, Type SE service-entrance cable used for interior
wiring shall comply with the installation requirements of Part II
of Article 334, excluding 334.80.
For Type SE cable with ungrounded conductor sizes 10 AWG
and smaller, where installed in thermal insulation, the ampacity
shall be in accordance with 60°C (140°F) conductor temperature rating. The maximum conductor temperature rating shall
be permitted to be used for ampacity adjustment and correction purposes, if the final derated ampacity does not exceed
that for a 60°C (140°F) rated conductor.
Since you used mechanical setscrew-type (Polaris™) splices for your tap, you should be fine with the copper-to-aluminum connections as those connectors are Cu7Al or Cu9Al rated in their common incarnations. Furthermore, the 83% rule for the run wires from the meter-main to your unit still holds thanks to 310.15(B)(7) point 3:
(3) In no case shall a feeder for an individual dwelling unit
be required to have an ampacity greater than that specified in 310.15(B)(7)(1) or (2).
Finally, the formal feeder tap rules do not apply here at all since the conductors from the junction box to the new subpanel are of a size which is adequately protected by the feeder breaker (#2 copper wires in conduit or a SE cable can handle 115A) without further consideration. (In other words, they are not tap conductors in the Code sense of the term -- they would need to be too small to handle 100A in order to be considered as such.)
...and the subpanel in the garage
The installation of the receptacles and subpanel in the garage is close to correct, but needs rectification on a couple of points. The good news is that these are not hard fixes, and should be manageable in a day -- nothing major was screwed up here, as you appear to have your neutrals and grounds sorted, and even though this box is full, that's an acceptable concession to make for something special-purpose.
The first issue is that the 50A breaker is a bit large for 8AWG wire going to a receptacle -- it would be fine if that 8AWG was going to say subpanel lugs via a conduit, as you could use the 75°C column in that case, but most receptacles are restricted to 60°C service, so that limits you to using the 60°C column in the ampacity table for circuits feeding receptacles, no matter the type of occupancy. Swapping the 50A breaker for a MP240 or QP240, presuming that's the correct type for what appears from the interior layout to be a Siemens/Murray spa panel, will correct that, while the NEMA 14-50 receptacle can stay as NEC Table 210.21(B)(3) permits 50A receptacles on 40A branch circuits. (It also permits 40A receptacles, but there are no 40A NEMA receptacle configurations.)
The other thing I noticed in your picture is a panoply of utterly redundant grounding wires, at least if the NEC grounding rules govern unchanged in your jurisdiction -- the RMC nipples are an adequate grounding path from the tap box to the subpanel and from the subpanel to the junction boxes in this case (i.e. not a service, and less than 250V to ground), meaning you simply need to use appropriate gauge ground pigtails from the receptacle to the box to finish the grounding path, given that you are using "bump out" covers on your junction boxes. Furthermore, the bonding bushing where the feeder enters the subpanel shouldn't be necessary, either -- that gets rid of the wire from the bonding bushing to the ground bar. In the end upshot, your ground bar will be rather empty, but that is fine given that you are using metal boxes and metal conduit.
Adding the stove is fine too, once you send all the Zinsco hardware to the smelter
It turns out that adding the stove only adds 16A of factored load, maximum (40% of 40A), to your 220.82 calculation, which puts you at 87.3A of computed service load, which is still within what a 100A service can handle -- the electrician who says you absolutely need a larger feeder to handle the additional load is incorrect. However, you might as well get a new meter-main fitted anyway (with the cooperation of the other stakeholders), as you have a serious problem: not only is your existing subpanel a Zinsco, your existing meter main uses Zinsco breakers as well! This renders this entire electrical installation hazardously unserviceable, as a documented failure mode for Zinsco breakers is them not turning the power off when you place the handle in the OFF position.
At this point, what you do is up to you and your fellow stakeholders. A two-socket, 200A unit of whatever specification your utility accepts is not a bad choice for the replacement meter-main as it's one less piece of hardware to swap if future expansion is called for, while I would fit a 42 space minimum, 200 or 225A main lug panel with ground bars to replace the existing subpanel pyromaniac monster lurking in your closet.
Best Answer
Yes, that's what it means. However you get a reprieve on large appliances.
You have 3 types of wiring generally.
My general thinking is that #2 aluminum is about the same price as #10 copper.
They are saying feeder can't be aluminum unless it's #2 or larger, which is fine by me given the above price point.
Also that aluminum is banned for branch circuits of any size; for instance wiring to an 80A EVSE must be #4Cu not #2Al. But here's the trick. EVSEs and most other very large appliances need disconnect switches. A 4-space subpanel qualifies as both a disconnect and a subpanel, turning the long run into feeder. Thus the #4Cu to an 80A EVSE only needs to be 2 feet long. The rest can be AL.
Likewise, a 60A hot tub can be run with #2AL (overkill) to a "hot tub subpanel" there and then copper to the tub proper.
There is nothing wrong with aluminum heavy feeder; it has always been reliable - but then, heavy feeder is typically torqued properly and used on terminals actually rated for aluminum. There was a problem in the 1970s when aluminum wire was used for small branch circuit wiring, however those terminals were not properly rated for aluminum, and nobody was using torque screwdrivers for the small stuff. (we only learned of the importance of that about 10 years ago).