I don't think that you're being obtuse. But I am beginning to suspect that you're having a good time at my expense: "Let's see how long I can keep Bill talking about infinity, how long after he's made himself perfectly clear on all the issues I've raised." If this is a hoax of yours, I wish you'd stop it! I care about education, and I've been perfectly sincere so far. It's getting harder and harder to imagine that you have been too.
I will give you the succinct answer you asked for when I finish defining things correctly.
***************************************
Any set S is a SUBSET of a set S' iff (if and only if) there is no member of S foreign to S'. (So every set S a subset of itself; and the set with no members, the empty set, is a subset of every set.) A set S is a PROPER SUBSET of another set S' iff S is a subset of S', but S is not the same set as S'.
Two sets S and S' are the SAME SIZE iff there is a 1-to-1 correspondence between their members: each member of each set has a "mate" corresponding to it in the other set, with no members of either set "left over" unmatched.
The NATURAL NUMBERS are 0, 1, 2, 3, ... . A set S is FINITE iff its size is given by one of the natural numbers: i.e., iff S has 0 members or 1 member or 2 members or 3 members or ... . The infinitude of a set can be defined in two equivalent ways. A set is infinite iff it is not finite. Or, a set is INFINITE iff it is the same size as at least one of its own proper subsets. The set of positive integers {1,2,3,...} is obviously a proper subset of the set of natural numbers; and it is the same size as the set of natural numbers (since with each member n of the set of natural numbers there can be paired the member n+1 of the set of positive integers, and with each member m of the set of the set of positive integers there can be paired the member m-1 of the set of natural numbers). So the set of natural numbers is infinite, since it is the same size as at least one of its own proper subsets: the set of positive integers.
Now a set is DENUMERABLE iff it is the same size as the set of positive integers (or, if you prefer, the same size as the set of natural numbers). So these are all denumerable sets: the set of even positive integers, the set of odd positive integers, the set of
prime numbers. It may not be obvious, but these are also denumerable: the set of all positive and negative integers, together with 0; the set of all positive rational numbers; the set of all positive and negative rational numbers; a set whose members are all the members of denumerably many denumerably large sets, lumped together; and so on.
A set is COUNTABLE iff it is either finite or denumerable. (A set is sometimes said to be COUNTABLY INFINITE iff it is denumerable.) A set is UNCOUNTABLE iff it is neither finite nor denumerable.
One set S is said to be LARGER THAN another set S' iff S' is not the same size as S, but S' is the same size as a proper subset of S. S is SMALLER THAN S' iff S' is LARGER THAN S. It follows that any uncountable set is larger than any denumerable set. (But the set of natural numbers, though it has one more member than the set of positive integers, is not larger than the set of positive integers: we've already seen that, and why, they're the same size.
*************************************
Now, let me answer your final question as briefly as possible. A set is finite iff it has no members or 1 member or 2 members or 3 members or so on for all the natural numbers. It is INFINITE iff it is not finite. It is DENUMERABLE iff it has exactly as many members as there are natural numbers. (So all denumerable sets are infinite, but not all infinite sets are denumerable. There are more than denumerably many points on a one-inch line, more than denumerably many different sets of natural numbers, more than denumerably many real numbers from 0 through 1; more than denumerably many distinct denumerably long strings of T's and F's; etc.)