How to determine where lead paint was used

The paint/primer should be fine, especially with two coats, but I typically begin with primer, as it seems (in my experience) that primer holds paint better and lasts a bit longer before requiring touch ups or repainting, and it's often cheaper to do a coat of good primer (match latex to latex, or make sure you follow the label if you use a different combo), then two or three even coats of less-expensive paint (paint+primer is almost always 2-3x more expensive than paint alone.

I wouldn't sand and re-paint, of course. You should be fine with what you have so far.

[Edit: one other note on paint+primer paints; make sure you read reviews or ask others about their experiences with the particular brand you're buying. A friend ended up with a patchy-looking paint job when he used one of the cheaper brands. I don't know if it was an isolated incident or if the brand is normally bad (or if he didn't mix the paint well!), but research before you buy.]

If the prior owner used caulk, and they were really dumb and didn't use paintable latex caulk, then its possible that this would be the result.

Regardless, obviously the paint isn't taking properly here - so yes, the answer here is get a quart of primer go over the edges with a brush, let it dry, and redo the corners.