p_S = \prod_{x \in S} x
And then later find out that the set S contains one element. Or it might even be the empty set in which case you'd interpret the above as equal to 1. This makes it work that
p_{S union S'} = p_S p_S'
in all cases.
When the sequence has one element, then that element is the product. So, 2 is the product of (2).
When the sequence is empty, the product is 1.
Set_1 includes 1, Set_2 does not.
We have two possibilities for naming.
Option A: Set_1: Primes. Set_2: Primes except for 1.
Option B: Set_1: Primes and 1. Set_2: Primes.
You can then show the differences in the properties that make having Set_2 have the default name an easier to use convention than letting Set_1 have the default name. But at the core is the understanding that this is an arbitrary distinction done just for ease of use. Had we called Set_1 the Primes, or if ever someone became Galactic Emperor for a day and made the change, then Set_2 would get a new name for ease of use and we would just use it instead.
There are many other sets that are really close as well, which have their own properties.
Side note, SET_1 is listed on oeis https://oeis.org/A008578 and still includes 1. It is called "Prime numbers at the beginning of the 20th century".
I assume most people on this list are engineers. They have college level math but not necessarily number theory. Number theory requires a similar thinking to the problems we solve in all the time in engineering fields.
I just explained the unique product of primes argument to my mechanical engineering husband. He got it right away and also thought it was awesome!
"My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way. If 1 were prime, we would lose that uniqueness. We could write 2 as 1×2, or 1×1×2, or 1594827×2. Excluding 1 from the primes smooths that out."