There will doubtless be a glut of more informed (and less informed) replies, so I will simply say this.
Labels are often wrongly ascribed, and I would venture to say that anybody referring to your family as MO would be shooting wide of the mark.
And it is not as if there is any concrete meaning to any of these terms, they depend on people’s perceptions, which can be very misguided. There is a significant difference between being from a ‘normal orthodox/heimish’ but with a more modern bent, and being a self-identifying MO. And, as in most things, it is not black and white, but on a spectrum with many shades of grey.
But with regards to what most people would call MO, it is where the ‘modernity’ impacts significantly on their understanding and practice of yiddishkeit, which can get very problematic. This can be found amongst certain ‘official’ MO organizations and movements, where interpretations and hashkafa get skewed. I am not going to make any judgements about the extent or severity of this issue.
So to sum up, there is a world of difference between Modernly Orthodox and and Orthodoxly Modern. Especially since it is something of an oxymoron. And I am being deliberately soft, partly to avoid causing offence and partly because I do not claim to have a deeper knowledge of the subject matter, although if I had the time or inclination to delve further I suspect I would be much harsher.