I know "invagination" from reproductive biology. The fertilised egg divides repeatedly until it becomes a solid ball of cells (a morula, latin for mulberry). The morula cells continue to divide to form a hollow ball (a blastula, latin for bud). The blastula then starts to invaginate in one spot, and the outside cells migrate inside to make a hollow ball (a gastrula, greek for belly). So in my mind the "in" is because it's a movement, a becoming.

If you go back to latin and greek, the etymology of lots of medical words can be revealing. "Vagina" was latin for scabbard. (Latin had a different word for cunt, but I don't know what it was.) Medical folks in the 17th century thought it would be a clever name for the birth canal. The metaphor is pretty clear there, but it could be used as a sort of in-joke in front of women patients because women weren't taught latin.

There's no sword involved in a blastula becoming a gastrula though, so the metaphor must be the birth canal. A second-level metaphor, I guess.

A recent domestic event caused me to look up "epididymis," which turns out to be greek for "near the twins."