Eisteddfod is a compound word (second word soft mutates)
… the fod might be from “bod” but not the usual meaning “to be” … Bod is also an old Welsh word for a permanent residence etc…
Eisteddfod could also just mean a place of sitting…as someone is crowned and seated on a chair
Secondly … a Cae is a field … but maes can be an expanse of open land usually enclosed (and not a wooded tract of land!)… you see the name used sometimes for battlefields … also maes parcio - open area for cars parking!
I always taken as a kind of “ini” in english (say it quickly). The same as “yndefe”, “ife”, and various others from elsewhere. An almost meaningless positive tag.
When I’m looking up a word to find out the Welsh for it, I generally check the word that I find in the other side of the dictionary to be sure of what sort of whatever-the-word-was it is. Especially when the dictionary in question doesn’t give that sort of hint in the first place.
For example, if I had looked up “lock” in the English to Welsh side, and gotten that list, I would then have looked each one up in the Welsh to English side to get the finer distinctions. It is very helpful!