What does "Isreal's right to exist" mean?

Does it mean that, given that Israel is here, that it should have a right to remain"?

Or does it mean that Israel has a Biblical right to confiscate all land historically linked to Israel?

These are two very different things.

The word "antisemitic" is not only used in a literally incorrect way, but it is used as a weapon to silence any criticism of current or past Israeli policies and actions.

There is also the issue of identity - people claiming that they are inherently different than others simply because of a theoretical birthright. In other contexts we would view that very negatively.

We need to differentiate these different ideas. Every person has a right to exist. And no person is inherently different than any other, no matter who they claim that their ancestors were. No person has any special innate rights by birth. Any assumed birthright is an artificial human construct, or is imagined based on superstition.