Two things are needed to create an s-trap out of a p-trap that connects to a vertical drain pipe:
a) short trap arm (less than 2x its pipe diameter)
b) lack of vent
If you only have [A], then it’s a crown vent, also not allowed in most areas.
If you only have [B], then it’s a p-trap without a vent, not an s-trap.
Anything that does not fit exactly A&B I just call improper p-trap because there are things like, for example, a vertical leg drain. And so, to keep things less technically confusing and or open to debate, I just simplify it. Let the plumber call it whatever the heck he wants 