You still aren't understanding, because nothing I said involved "only within the sphere." It's "only within the sphere" is does teleportation fail. It does not fail outside of the sphere.
Teleportation moves you from one place to another without crossing the intermediate space (in particular because you do not choose the path that you take when teleporting). Because teleportation fails within the sphere, you cannot appear there, full stop.
So as written, because it does not in any of your quotes cause the spell to fail, the teleport from outside the sphere succeeds as a departure point and then fails as a destination point WITHIN the sphere.
And as it fails as a destination point within the sphere, it simply fails. You have no destination, no path, so you simply don't move.
Right. So the caster successfully departs from outside the sphere due to the teleport spell not failing, and then nothing. It fails inside the sphere as a destination and there are no rules for where the caster ends up.
The teleportation fails. The caster does not end up anywhere, because he is not moving along any path. There is only a departure and arrival point. Since there is no arrival, there is simply no departure.
Teleport doesn't fold space.
First, no-one says it does. But you would have a difficult time proving that it does not, try it. Show me where it says that it does not fold space.
The fact that other spells do something is certainly not a proof that another spell does not do the same thing. It's actually more an indication that it does.
Moreover, a gate spell does not really fold space, since it is also between planes.
Teleport just transports you, which means you do cross the intervening space, but you do so instantly.
Prove it. Show me where it says that you cross the space. It does not, and in particular, there is no choice of a path to cross it. This is 5e, and spells only do what they say they do. Does the spell say that you "instantly cross the intervening space"? No, all it says is that it instantly transports you. There is no mention of intervening space.
It does have two ends like a string, and like any string you can leave one end and travel to the to the other end, even if the other end is cut short.
There is no cutting. There is no path. Show me where the spell says that there is a path. There is ONLY a departure and a destination. If there is no destination, there is no teleportation, at all. As said in antimagic field, the teleportation fails.
Teleport works like that as written. You leave the destination successfully, since the spell works.
Antimagic field does not mention a spell. It just says that teleportation (not teleport, the spell) fails in the sphere when the destination is in the sphere.
You travel along the string instantly.
Prove it. Show me where the string is in the spell description.
And then the teleportation fails as a destination inside the sphere. So where does the caster emerge?
He does not move. There is no path, no string, just a departure point and a destination point, since that one is within an antimagic field, the teleportation fails, and the spell, although successfully cast, fails. Again, very clear in Xanathar in the section about invalid spell targets. Your target destination is invalid, since an antimagic field causes teleportation to FAIL when the destination is within the field. So "nothing happens to that target, but if you used a spell slot to cast the spell, the slot is still expended."