You are correct. It is a common error to refer to the lead as the pitch (i.e. “8mm pitch” when it’s not really 8mm pitch), so you often have to infer what is really meant, which might not technically match what is said.
As long as the lead is correct, the pitch does not really matter as long as the nut and screw match each other, but 2mm pitch is by far the most common so that’s what we all use.
The combo of pitch, lead, and starts gives you the info, you need two of three to get the right one (I sell 4 start 2mm pitch=8mm lead). After that it is all screw geometry (diameter and thread shape).