This isn't a new change (and does not coincide with the release of version 2). I've seen it called MBAM for at least a couple of years. MDOP and presumably MBAM itself goes back to Vista days (a quick check reveals 1st April 2008 as launch date), and whilst I can find no evidence of the MBAM acronym in use at that time, or even a mention of the existence of that time, I wouldn't be too surprised if it made it into the September 15th R2 - after all, BitLocker has existed since Vista.
A quick check of Malwarebytes suggests its MBAM was first released January 2008, and whilst I don't know how long it took to take off in a large way, it is conceivable that Microsoft decided on their name before Malwarebytes did, or before it became large. After all, Vista was first announced in July 2005 and Bitlocker must have been announced considerably before release too (although I can't find a specific date), therefore it is possible that MBAM was initially planned at a similar time.
So...Microsoft MBAM may well have been planned before Malwarebytes' AM. Just because it's a security tool, and this is in large part a security forum, doesn't necessary mean it came first.
I'll keep exploring which MDOP MBAM was first in, and when it was first announced, as this is crucial to this argument...