Ordinary EMT connector to a bonding bushing.
The basic common EMT connector you might use to go into a standard knockout (heck, you have one right on that existing box). I prefer the compression type especially if you might need to spin it in close quarters.
That connector has a thread on it, right? Enter the Tesla enclosure and screw a bonding bushing into it. Or if clearance requires, screw the connector into a bonding bushing! (remember what I said about "easier to spin").
If the Tesla has a 1/2" knockout you can go right into that. If it has a larger KO then you use a "conduit reducing washer".
Given the small quantity you need, consider a family owned hardware store (such as an ACE or HWI affiliate). They're likely to be more happy to sell you single parts at sane cost instead of making you buy a bag of 10 like the big-box stores do. I know my local "boutique" hardware store does. For the more obscure stuff, an electrical supply house.
Other thoughts
Most people think they need much more charging power than they really do. Here is a sanity check and good primer on the subject. If ya got it flaunt it, but if you don't got it, don't set your panel on fire flaunting it. OK?
Save your old hardware. Before you sell your house, roll it back to plug and socket. People make better offers on houses that are obviously set up for EV charging, and "14-50 socket" has more curb appeal than "empty hole were EVSE was".
Also, consider multiple EVs. Now, a large circuit can let 2 or even 4 EVs share a single current allocation dynamically. When you have multiple cars, a funny thing happens. It's exceedingly rare for them all to need a big charge. You'll have 3 cars, one needs 15 miles, one needs 30 miles and the other needs 180 miles. A 40A circuit will do all 3 cars in 11 hours dynamically on the fly, no clock watching, no swapping cords. Really.
6 AWG copper is 65A wire. Due to technicalities and Tesla not bothering to read NEC before designing their units, you must use no larger than the 60A setting when commissioning the EVSE.
Even if you didn't before, do a NEC Article 220 Load Calculation on your house's service excluding the EVSE. This is a Code requirement for additional loads. Take your service/main breaker size minus your Load Calculation and that gives you the headroom on your service. Round down to the next 5A increment and use that breaker size when commissioning the Tesla Wall Connector. Most people ignore this altogether and try to set their panel on fire. Insurance won't pay when you do that.
If you're really in a jam and just don't have the service to support the charge rate you'd really like, it's possible to use "load shedding" to solve this. Use a contactor or EVSE feature to have the EVSE interrupt (or simply slow) when other loads are running. The best EVSEs install a current transformer on your service wires so they dynamically change as loads come on/off.
I'm less familiar with EMT so would likely use PVC conduit
You'd be better off seizing the learning opportunity and learning about EMT. It really is marvelous stuff, doesn't cost any more than PVC, performs better, and best of all, errors are cheap since everything unfastens. The most you'll ever waste is a little bit of stick. Unlike PVC which must be glued.
Note that the conduit is really long and curvy, so it's not practical to try and feed another conductor through it all the way to the breaker box.
Another thing worth skilling up. Not that hard, the trick is, pull out all the wires and use them to pull in a stout pulling string, then rearrange the wires and pull it all back in. Make full use of any "pulling points" between here and there. If one of the "pulling points" is a junction box, you can grab ground at that intermediate box and don't have to go back to the main panel.
If there is somewhere unobtrusive yet accessible along the EMT run, you can pull the wires back beyond that point, and insert a common 4x4 steel junction box there (about a buck) with two EMT connectors, and then grab your ground at that box.
Readers: You do not need 60A panel headroom (or really any headroom) to charge a car. See below. OP's valid, well-informed reasons are very unusual.
Wiring options
I think conduit is a great plan. For a first time learner, I recommend EMT. Why? Because it's an "Erector Set" - everything comes apart and there's no waste, nothing is glued so everything can be reused. It's also a viable grounding path, but not to a plastic case EVSE and we need a ground wire in the pipe anyway for conduit fill reasons.
I would use the green route, not the yellow route, because I disagree with opinions that Romex doesn't need protection there. It obviously does; the wall is being used to hang tools.
I wouldn't worry about hanging a junction box behind the EVSE. Just bring conduit in top or bottom. To step down from your conduit size to the EVSE's opening, just use conduit reducing washers e.g. this. The EVSE has a plastic box; it's not carrying ground so the reducing washer is just there to keep fingers out pretty much.
I wouldn't bother getting a tubing bender. I'd just buy pre-formed 90s, and LB, LL's or LR's (conduit bodies) for making turns too sharp for two sweeps back to back. A junction box (e.g. 4x4 box) also works as a conduit body - cheaper too.
99% of my work is EMT conduit, and I don't attempt compound bends, I just use simple bends and couplers. I fit pipe up to the wall, note where they overlap, mark with a sharpie and cut there with a mini hacksaw e.g. Stanley 20-807. (takes little space in my electric toolbox). Leaves a negligible burr easily removed with the multi-tool e.g. Klein 1010, that's why multi-tools are wedge shaped. Don't use a tubing cutter, ever - it'll take all day to ream out the huge burr.
I would use the green route, and use "sweeps" to make the turn from vertical wall to horizontal. Where horizontal meets horizontal, I'd use either an LB or just a 99 cent 4x4 junction box (cheapest conduit body). Wire count and box fill aren't a consideration if there are no splices.
Conduit must be built and anchored total and complete before any wires are pulled into it. This is much easier, believe you me and it also allows you to do the work in sections. I'd build up the wall from opposite ends and let the pipes tell me where they want to meet.
#6 THHN is 65A. Both can be black. It's technically possible for two #6 THHN to fit in 1/2" conduit with a #10 ground, but that will leave literally zero room for the signal cable if you want to use the power limiting feature with the current transformers. So I would go 3/4" conduit. NEC rules don't allow the signal cable to run outside the conduit for reasons I won't get into here.
Beating 20A
I’ve studied the ‘20A is sufficient’ debate but decided 60A will be best for my use case.
That might work for you, but I really do not want readers thinking "OMG I need 60A to charge an EV, and therefore I can't get an EV". For most, yes, EVs take a lot of power, but not THAT much! So 50-60A "is expensive, and may come with more headaches than you bargained for".
Now fairly unique to your situation, OP, is your locale's waiving permits for <=20A circuits. That's a loophole most people in your town could drive a Rivian through :) Most readers won't have a locality with such a rule.
Next headache: many power companies require you to file a form/permit with them too to run a high-powered EVSE. Again not an issue for little ones.
And our next stop under "headaches" is the NEC Article 220 Load Calculation. Yes, I know you say you have that all worked out, and I'm saying this mainly for the benefit of other readers. Honestly, 95% of EVers "freestyle their own private Load Calc method" and declare victory. Nope, only your AHJ (permit issuer/inspector) can do that - some use reasonable alternative calculations, particularly for EVs. Here is Sacramento's.
Notice in any legitimate Load Cal formula, how many loads get a major "discount" for load diversity - the various loads in a home are vanishingly unlikely to be maxed out at the same time. EVs do not get that - much the opposite, they are derated by 125%. Once the rest of the Load Calc is complete, the difference between that and your service size, multiplied by 0.8, is your maximum charge rate. Hold that thought.
Anyway, if your Load Calc gives you 60A of headroom, you're all set for 48A, which is the max on current Teslas but exceeds the ability of most cars. The car's limit is trending downward because the charger is actually on the car, and costs money.
If your Load Calc does not, then you really need to get realistic about 20A LOL just kidding, we'll do 60A anyway! By cheating. Certain EVSEs provide for "current transformers" to be added to the service wires. A module in the panel communicates with the EVSE and warns it when the service is about to be overloaded. The EVSE dials down charge rate on the fly when that happens. When the coincidence of loads that causes the overload ceases, the charge speed rolls right back up. This allows us to completely ignore the Load Calculation, at the cost of paying a few hundred more for the kit. The popular domestic option is the Wallbox Pulsar Plus, whose North American version supports up to 48A. Overseas it's the Myenergi Zappi. Yes, you'll need an adapter for a Tesla but it can just live on the cord.
It's a headache, like I said, but a very doable headache. Heck you could charge at (quite close to) 48A even on a 60A service. This is probably the wave of the future, and the reason we don't recommend it universally is limited availability/choice right now.
With great power comes great responsibility. The times of greatest strain on the grid are summer days from about 2 PM to midnight (A/C load is offset from solar generation because the structure of the building stores heat). This creates a "duck curve" which makes it particularly damaging to charge EVs from about 5PM to 11 PM. So don't.
Anyway, 10 years from now, the charger on the car will be shrinking as data shows it's rarely used productively. I expect AC level 2 charging will be limited to 24A on most cars. Luxury-minded people will get a "Home DC fast charger" with NO upper limit to charge rate (thanks to those current transformers). A 200A service could charge at 40kW (~160A) minus house loads.
Best Answer
The 48A Tesla charger requires a 240V 60A circuit. That's 2 conducting wires plus ground. There's no need or use for a neutral wire. Per the installation manual: "use minimum 6 AWG, 90° C-rated copper wire for conductors." The appropriate ground wire size is 10 AWG Cu. Dual-rated THHN/THWN-2 installed in conduit and attached under the joists would be suitable. These wires must always run in a raceway or conduit (unlike NM cable that you see stapled to joists in dry indoor locations).
For a 60 ft run I would stick with 6 AWG Cu. The one-way distance for a 2% voltage drop is more than 115 ft. Assuming 0.5 ohm per 1000 ft you will burn 70W in resistive losses or 0.6% of the transmitted power. There's no need to install 4 AWG Cu wire unless you anticipate a second charger and expect to run both chargers all night each night (not a common situation, but I don't know your commute).
For a DIY project I would use PVC-40 since you don't need special tools to work with it and per your description the installation is not subject to physical damage. If the conduit emerges from the ground or runs under a garden bed I would use PVC-80. Two 6 AWG conductors plus one 10 AWG ground will fit in 3/4" PVC-40 (24% fill). The same conduit can handle 4 AWG if you needed to upgrade down the road (34% fill).
A strict interpretation of code section 422.31(B) says you need a disconnecting means (line of sight or lockable). My local AHJ is not requiring this for a 240V 48A charger as implied by 625.43.
Others should comment on the wall penetration.