No it's not. Logic is built from axioms- things you take as true without proof. If you take the statement "This statement is not true", and start a proof, you can reach an contradiction, and using the idea of contradiction by proof you can prove that something you assumed isn't a true statement, and since the only thing we assumed was the axiom, we know the axiom isn't self consistent and therefore isn't a valid axiom. Same thing with the two statement version, etc. etc.
Don't know anything about a Tarskis proof, can't look at it now. I'll check it out later if no one else comments.
That's the point- Logic
is built from axioms, like "if a then b", "a", therefore "b". The thing was that the liar might show these basic assumptions about reality to be wrong.
Also, you're partly right in your second point- what you mentioned is known as indirect proof. This is where you assume x to be true, and if you get a contradiction, then x must be false. It also works in the opposite way; assume ~x (or "x is false") and if you get a contradiction, then you can conclude ~~x, aka "x is false is false", aka x is true.
Take, for example, "the sky is on the ground". Let's define sky as anything that is not on the ground. So, the sky is on the ground and
not on the ground. So, it is not the case that the sky is on the ground.
If the sky is not on the ground, then, well, it's not on the ground. No contradiction. So, it must not be on the ground, meaning it is false.
With the liar, though, if it is false then we have a contradiction, and if it's not false we have a contradiction. So, no matter what, we have a true contradiction.