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I think I painted mine in glossy enamel.

We do doors and trims in semi-gloss. It does have a shine to it, but I like the contrast against the eggshell that we use on our walls.

I would do semi-gloss for a door.

Honestly, when choosing paint for my bathrooms I couldn’t really tell the difference between the samples of semigloss and satin from Sherwin Williams.

I was told the semigloss is a little more durable but the satin goes on easier. I went with the semigloss, but the guy doing the painting had to prime any spots that were repaired because the semigloss did not stay looking good as it dried. (I was using the exact same color of the old primer and paint.)
When he talked to a professional painter friend of his, he said he always puts in his contracts that he will NOT use semigloss paint.

Semi gloss because it’s easier to clean and doors are notorious for getting dirty and fingerprinty.