Usage of Word "Exactly" In Quant Problems

As seen from your post itself, that would be the prime factor length

eg: 1080 = 2 x 2 x 2 x 3 x 3 x 3 x 5

The number of different prime factors would be 3 (2,3 and 5)
The prime factor length would be 7

And IMHO set A has two numbers means only that. i.e It has only two numbers. If it had said "at least two numbers" it would mean that set A has two or more numbers.

The usage can vary a little from question to question but in most cases the intended meaning is quite clear.

y has two prime factors
=
y has exactly two prime factors
=
y has two distinct prime factors
=
y has exactly two distinct prime factors (gives the meaning clearly so is most preferred)

'Set A has two members' means 'Set A has exactly two members and no more'

There is a source of confusion in this concept. Let's discuss that with an example.
There were 100 people at a party. The food at the party consisted of three things - cake, candy and corn.

'50 people ate cake'
This means exactly 50 people ate cake. The other 50 did not eat cake. The only thing is that of these 50 people, some or all could have eaten something else too. This statement doesn't imply that these 50 did not eat anything else. You have to remember this.

'50 people ate only cake'
This is different. This means 50 ate cake and nothing else.

50 people belong to group1 means exactly 50 belong to group1 but the point is that they could belong to other groups too.