Oh no, a whole new bunch of time-wimey words I hadn't even considered! :-)

I'm no authority but to me using "the other day" as a phrase is trying to impart a rough reference to a point in time.

Which means we're talking in "day" timeframes where week or month wouldn't be suitable. I would expect "the other day" to be within a week, but accept no more than a month.

Similarly "some months ago" = probably less than 6 months but definitely no more than a year.

The world is a wonderful place and the ways we humans can confuse communication seemingly knows no bounds. But I don't want a proscriptive definition for each little casual phrase as I think this flexibility in language is why it keeps changing and being alive!

I'd claim that the time ranges for these depend on what we're talking about.

"I had a sandwich a while ago" is wastly different from "they had a child a while ago".

"The other day" and "back in the day" might not differ as much but still...