Can't tell from the pics whether this is showing a badly-applied paint job or something coming through the finish?

What color primer did you use?

It is a new door that has never been painted. You still see the wood thru the paint even after 7 coats. The primer was yellow, like the paint color. It is an eggshell finish.

What brand of pAint is it ?

This is the inside of an exterior metal door with 2 coats of Valspar Signature Semi-gloss paint. The door was previously unpainted. I like and agree semi-gloss is the best choice.

If you really want a professional look without any wood grain first you should of primed it a oil base primer tinted to yellow then filled the grain with a filler sanded it down and primed it again then top coat the door with Benjamin Moore grand entrance in high gloss