[nyeah]

I've often heard "Irish" for Irish and "Gaelic" for the Scottish one. Is that not used systematically?

[closewith]

Yes, Irish in called Irish in English and never Gaelic, which is used as an adjective. For example, the Gaelic Athletic Association.

In Irish, Irish is Gaeilge.

[Asraelite]

A lot of people call Irish "Gaelic" colloquially. I hear it especially often from Americans. No idea how common it is in more formal settings.