The sight is one thing with three aspects. The two g33k pointed out and soul gazes.
I'm willing to believe that a Soulgaze is related; but it isn't the same.
You can only have it once per person; it begins, happens, and ends. If you don't avert your gaze, it is unavoidable.
It is something mutual, so even if the other party has no Sight, they still join in... just as if they had initiated it themselves, and did have the Sight.
These each seem to be different things. I'm willing to believe the three are related, to believe they have some common causal feature. But I'm not convinced of it; I don't think we have in-'Verse canon, or WoJ.
As you say -- it doesn't take a Wizard to have "the Sight," the true-seeing aspect of it. But IIRC you do have to be a Wizard to do a Soulgaze; not just anyone with Sight can do that.
I'm unclear on the involuntary-precog thing... it's something that happens after a while of wizarding, but can it happen to others? It's not quite Cassandra's Tears... but it's not entirely dissimilar, either.