No, you can't be both LADA and MODY, I never suggested you could. The way my doctor described it, I'm not a type 1 yet though obviously headed in that direction, because I have a greater degree of function remaining than a classic type 1. Historically there were only the two types (sorry, three if you count gestational) of diabetes but researchers and the medical profession are realising that not everyone fits neatly into those boxes and that other options need to be considered hence the recent proliferation of labels. Maybe they'll decide in the end that there's just type 1 and type 2, or maybe the classifications will expand even more as medical knowledge increases. I have no problem being labelled LADA if it fits my personal version of the disease and means I have a better chance of getting a good level of control as my treatment regime is adjusted to fit my needs.
As for going onto insulin, we discussed it and the idea terrifies me, silly though that may be and I want to give the meds a fair chance first. It's an emotional not a logical response and it'll take me a while to move past that. The new pills may yet do the job, it's only been a day or two so far, I'll see where I am in three months time and then decide.