No, an LBA in general doesn't have a finite tape. It still has an infinite tape, to accommodate arbitrary length inputs, it's just that the tape cannot grow beyond the length of its input (or a constant multiple of it, which is equivalent by the linear speedup trick).
No they really are. Arbitrarily long means "pick a number any number." Infinite means, well infinite. And infinity is always bigger than "pick a number, any number."
I suggest that you read Michael Sipser's Introduction to the Theory of Computation. It's absolutely lovely and an easy read, considering the subject matter. It will help you to understand cardinality (and many other things) in a pragmatic computing science context.
It may be surprising to you that I've actually read a good portion of that text as well as other books on the subject. Or that I actually do understand what a cardinality is in a "pragmatic computing science context" (whatever that's supposed to mean).
> Infinite means, well infinite
There's a technical definition of "infinite cardinality" which is a bit more rigorous than "well infinite". If |S| > n for every natural number n, then S has infinite cardinality basically by definition.
Why don't you try implementing an LBA simulator in your favourite programming language with a finite-sized array and tell me how it goes? Anyway, the original point was that a Turing Machine with a finite tape is an FSM. This is true (well, or more technically: it's equivalent to an FSM) because you can encode all possible configurations of the tape with a finite set of states and thus don't need any extra memory.
