Is picante sauce a type of salsa?
There's not really a definition as precise as you're looking for. Picante sauce refers to either Pace or something similar, depending on the person. So yes, picante sauce is a somewhat ill-defined type of salsa - nothing that people call picante is not a salsa. As you point out, it does not actually mean intense heat; the regular/medium is not particularly hot and even the "hot" isn't all that hot, and people will refer to any variety of Pace Picante (or similar) as picante.

Pace's distinction between the two is more of a branding/marketing issue than anything else. They have to keep making picante, without changing the recipe too much and certainly without changing the name. They also want to make other things, and salsa is a pretty natural name. That doesn't mean their Picante isn't a kind of salsa, just that they're choosing not to market it as one. But as long as you consider basic mass-produced jarred salsa to still be salsa, Pace Picante is definitely an example of it.

The reason there's no satisfying answer or precise definition is that all usage of the term like this stems from Pace Picante sauce. It's just the name someone chose for their sauce in 1947, and since that sauce eventually became widespread in stores across the US - stores that likely didn't have a lot of other salsa - people ended up using the term in a more casual, generic way.