I recently worked a crossword that had to have been written by a Brit. Answers included "have at it", "anorak" and "paint pot". Nobody I know on this side of the pond says paint pot. Whether the lid screws on or flips up, we say bottle. If it comes in a tube, we say tube. If we're not sure what to call it we say container. Say pot, at least in Northern California, and folks certainly don't think of a paint container.
So what do you call this?
For those unfamiliar with such a thing, what you are looking at is a contact lens case. This one has lids that screw on and off. The older kind used to have lids that were attached and snapped into place. They wore out, leaked, and had the potential to send your precious lens flying if opened incorrectly.
This version shown here makes a great little temporary storage for paint. I have mixed a custom color of three drops white to one drop "Butter Cream" and a drop of matte medium thinner to match the aging sides of the MDT freezer, in order to paint the ends. When you just need a tiny little container to keep a custom color fresh long enough to give the object you are painting multiple coats, this fits the bill nicely.
Whether you call it a pot, container, vessel, or whatever, I call it really useful!