I think the usual abstract algebra definition is one factor: itself. 1 is not considered a factor so that prime factorizations are unique, otherwise you could tack on an infinity of ones.
The abstract algebra definition is that p is a prime if whenever p divides a product ab then p divides at least one of a and b.
Having no proper factors is the definition of an irreducible element.
The two definitions agree for the integers and nice algebraic structure (UFDs) but there are algebraic structures in which they are not the same which explains why there are two differenr definitions and names in abstract algebra
Pardon my extreme ignorance of the subject, but could you elaborate on why your statement is mutually exclusive of the OP? Not at all combative, just a sincerely interested layman :)
Define "easier" in this context. If you have an algorithm whose complexity increases exponentially with the key length, saying you made a 10% gain doesn't mean squat.
Primes by definition have two, and only two factors.
The difficulty of factoring a number which is the product of two large primes is the important bit.