[v9] Remove `Client` type, rename `BaseClient` -> `Client` and use that instead

[mydea]

Currently, we have an abstract Client type in @sentry/types, and the BaseClient in core which implements this. However, since we use the Client type more or less everywhere, whenever we add something to the BaseClient we have to guard it because a user may have a custom Client that does not extend from the BaseClient (in theory), which bloats up our code.

So in v8, we should get rid of the abstract Client type and instead just make sure that every client always has to extend from BaseClient (then just Client), which means that we can add things to the Client more easily going forward.

[mydea]

Supercedes #7727

[AbhiPrasad]

This is really nice to have, but we realistically don't have the time for this, moving to v9 milestone.

[lforst]

This strongly depends on #14271