Apparently, in Middle Welsh you could sometimes put a pronoun before the verb, for emphasis -- as in mi welais, ti welaist etc. for welais i, welaist ti etc. (Fe/ef was the equivalent of modern e/o.) So mi was originally just for talking about yourself (because it really did mean mi) -- but then again, fe was just for 'he'.
But Welsh sentence structure shifted round to normally having the verb second, generally with some sort of particle before it -- like yr giving us all those literary-sounding rydw i etc. forms from yr ydwyf i, or na(d) for negatives, like nad oes -> does. And so mi and fe on the front of the verb got kind of re-interpreted as pre-verbal particles, and lost their link to being just for one person -- according to what you read, mi got generalized in the North, and fe in the South. So it's interesting that the South still feels that mi sounds like you mean 'I' -- I have to say it trips me up half the time when I hear it, and my brain goes from Mi oedden nhw to "I was - no, wait a minute, hang on - they were..."
Not that this helps with your invitations, mind.