The door of durin.

[Durelin]

In answer to your question, Eowyn Skywalker: Because Celebrimbor was an elf, and so he would use elvish no matter how close a friend he was to any dwarf. And most likely no elf and dwarf were close at all until Legolas and Gimli. And elves were the only ones who knew the skill of writing the script in the 'moon-letters', I believe. Another answer to that question can be found in Voralphion's post:

The doors were written in elvish because the western door was the one mostly used by the elves of Hollin when the dwarves and elves were friendly.

It could be that Moria was simply a mistake made by Tolkien. It must have been nearly impossible to keep up with an entire mythological history, including as small of details as the changing of names over time. But it could also have many other purposes. For one, Moria could always have been Moria, though the darkness of its name would simply be descriptive, and would not have to signify fear of it. Perhaps it was inscribed so to foreshadow later events, and because of these later events was adopted as the common name for it.