The rules imply that objects have two different "modes" of existence. They can be independent objects, subject to targeting and damage from area-effect spells. Or, if carried by a character, they become "attributes" or "parts" of that character, and no more subject to specific damage than, say, the character's ears. These are not mutually exclusive - an object can be carried by a character and still be targeted by a spell or action. But they cannot be both at once in the context of a single action.
Note that it is the attacker, not the character, that determines which "mode" applies. The attacker indicates this by specifying what he is "targeting".
In the OP example, the objects carried by the victim would not be subject to the spell effects unless either 1) the caster targeted the specific item or 2) the spell specifically indicates that carried items are affected.
For Banishment (or Shove), since the carried items were not specifically targeted, they are considered to be part of the character and are banished with the character. There is nothing weird about this. After all, hair is nothing more than dead keratin that happens to still be attached to the skin, yet we would consider it even weirder if Banishment left the victim with a bald pate and silky-smooth legs. We intuitively understand that a character is more than just his living tissue, and the game makes no distinction between items that are carried in a pouch, gripped tightly in the hand, or attached like hair to the body.
The Pinocchio Paradox
There is one strange glitch in the game's handling of objects which I call the "Pinocchio Paradox". Suppose Geppetto is playing with his marionette when his archenemy strikes him with an area-effect spell. The puppet, being "carried" by Geppetto, is unharmed. On the next turn, the Blue Fairy turns the puppet into a real boy, and then Geppetto is struck with the same area-effect spell. This time, Pinocchio is caught in the spell and takes damage.
Although at first this seems like a hole in the rules, it really isn't. The essential distinction between the puppet and the boy is that the boy can take independent action. Damage in the game is only recorded when it has a game effect - that is, it affects the types of actions possible and the probability of specific results. On the first round, of course the puppet was damaged in the colloquial sense - the DM might describe the charring of the wood or the tangling of the strings - but that damage has no impact on combat and therefore is not recorded. On the second round, the damage done to the boy does have an impact on combat, as the boy has the potential for independent action which he may or may not now be able to (or wish to) take advantage of.
You can further imagine that the puppet could be magically (or mechanically) animated to make the situation seem even more paradoxical, but the rule for resolving it does not change. You can also consider the opposite paradox - is a baby being carried count as an object or a character? There is a simple test for these situations - is the object (potentially) capable of taking Actions and is therefore a "character"? If not, any damage (unless the object is specifically targeted) is considered cosmetic and for simplicity's sake is not recorded.
Option 1: Once Damaged, Twice Healed
Assuming Life Transference is valid for use with Twinned spell (see below), the key phrasing that leads to this outcome is the following from the Twinned Spell description:
...you can spend a number of sorcery points equal to the spell's level to target a second creature in range with the same spell.
This specifies that we don't cast the spell twice, but rather the effect of the same spell applies to two targets instead of one.
The relevant part of Life Transference is:
You take 4d8 necrotic damage, and one creature of your choice that you can see within range regains a number of hit points equal to twice the necrotic damage you take.
The amount the target gains is equal to what you lost. Splitting wouldn't meet the effect of having each target gain an amount equal to what you lost. It's worth noting that spells that deal damage aren't split either, so it shouldn't for healing.
However...
There's an argument to be made that Life Transference targets both the creature and yourself, as you pointed out yourself in option 3. Under this interpretation, Life Transference would not be a valid spell for use with Twinned Spell.
Best Answer
D&D 5e is a bit ambiguous with its terms, so there is probably never going to be a comprehensive answer
The linked podcast is probably the best source of Rules as Intended we'll get. It says that "target" should be used in its natural English language meaning. This means that anything affected by a spell can be considered a target. I'm going to use lower-case target for this kind of meaning in the rest of this post.
The other main meaning is "the thing you choose as a target" according to some rule (such as a spell's text). This is a semi-formalized meaning, and Allan Mills' answer quotes some of the rules for this kind of targeting for spells. But elsewhere, the formal terminology blends in to the other less formal meaning a lot. I'm going to use all-caps TARGET for this meaning.
The TARGET meaning is definitely not the only one used in the rules, as spells like Fireball have one TARGET you choose (a point in space), but may also effect a number of creatures (and flammable objects) in the area of effect. Affected creatures are are explicitly described as targets in the spell's rules (emphasis added):
Unfortunately, some spells make it hard to judge who or what all the targets are. As the linked question you've provided show, it's not very clear if Warding Bond or Life Transference (from Xanathar's Guide to Everything), which both have the capacity to damage the caster, should treat the caster as a target or not. By the the logic in the podcast, the caster is affected and so must be a target. But they're definitely not the chosen TARGET, and it's possible that some parts of some rules (like the one that terminates Warding Bond early if either character has the spell cast on them again) may only want to refer to the character being the chosen TARGET of a future casting.
Another area that is somewhat ambiguous is the destination of teleportation spells. As the formal rules for picking spell TARGETs say, you normally need "a clear path to the TARGET":
One surprising consequence of this (as the podcast discusses) is that you can't cast spells through glass windows, since they count as total cover (personally I'd interpret cover relative to the type of effect, so a window would not provide cover to a non-physical effect like Hold Person, but it would indeed cause a Fireball to explode prematurely, though the blast might destroy the window and let the AOE spread outside). But most teleportation spells are phrased such that you don't explicitly TARGET your destination, even though your arrival will clearly affect the place. So it's unclear if you need to a clear path to the destination or not. For some spells like Dimension Door and Teleport, it's pretty obvious that you're not expected to have a clear path, since they have specific rules for how you pick your destination even if you can only describe it or have had it described to you (an example given in the rules: "upward to the northwest at a 45- degree angle, 300 feet"). But other spells like Misty Step specifically say you need to see your destination, which makes it ambiguous if you can use the spell to teleport through transparent total cover (e.g. a window). The explicit TARGET of Misty Step is the caster (it has range self), but some spells are less carefully written, and so we find spells like Thunder Step (from XGtE) which specifies the distance you can teleport with its range, making a stronger case for the destination being a TARGET too.
Ultimately, these ambiguities are design problems that the creators of D&D have not yet fixed (and they may never do so). As such it's up to each DM to make a ruling for their own table any time they come up. It is a clear design principle of 5e that the DM should be empowered to interpret the rules as best works for their game, and that the rules do not attempt to be entirely comprehensive.