I know you got a lot of answers already, but I thought I'd put my two-cents in because I have lots of perspective! :-) anyway, my parents divorced when I was 3 and I gained a new bonus dad at 4 and when that happened, my mom told my brother and I that if the person acts like a dad and treats you as such, then they deserve the title of dad too. However, when at 7, I gained a bonus mom (I too hate the title step mom/dad because it indicates that they are something less, but they had to be in the thick of it too raising me), my mom must have forgotten that conversation with us because when it came to choosing what we would call her, we chose "mom" again...we were given the opportunity to either call her by her first name, "mom" or choose something else, but we were older by then and understood more than a 3/4 years old obviously. Boy was my mom upset! She asked if we would call her by her first name around her and when I slipped up, she would get this hurt look on her face, so I grew up referring to her as "mom" at home (they had custody of us) and her first name around my bio mom...it was very stressful because I was always afraid I was goingto slip and disappoint my bio mom again...I was a people pleaser especially as a result of the divorce...I did that until I was grown...then I figured why? She's been such an integral part of my life, nearly as much as my own mom, why should I have to freak out each time I'm around my own mom about this that I started referring to her as mom around my bio mom again, though I think after this many years, she isn't as hurt by it, but let me tell you, it still makes my heart race when I'm talking to her about bonus mom (let me add...my bonus mom always stressed that she knew we had a mom and she was never going to replace her but I believe the title of mom helped us grow closer...plus my mom divorced my bonus dad and he's still in my life, still call him dad-which my bio dad always respected my decision to do that...I imagine it stung at first, but he put his feelings away and let us do what we needed as kids...and foremost, I never forgot who was my bio mom and dad in the mix...the titles just brought me closer to each one and reflected how I feel about them...when I talk to other people, I do try to clarify by saying "mom here", "mom in whatever state they were living in at the time...but they were still mom)
I do agree that if you want to try the I'm mommy, she's mama route, that would be perfectly acceptable, but discuss it with the other set of parents because she's so young and they will need to help reinforce it, but do it gentlywhich it sounds like you have lots of compassion for this woman who will now share in the burden in carrying for your daughter :-) you have a great head on your shoulders about the whole thing. I think feeling surprised and stung is a valid feeling...I would likely too.
Another thing is my bonus mom and dad both refer/introduce me as their daughter, not step daughter and that makes me feel amazing inside and accepted and wanted every time they do and my kids are their grandkids!
Okay, another perspective...I'm also a daycare provider and I can't tell you the number of my daycare kids that start off calling me mom (especially if I've been with them since birth) because that's what they hear my kids calling me...it flatters me (yes I correct them, but it still feels good)...and on the flip side, my daughter who just turned 2 took a stint calling me S....Not because she didn't know I was mom, but because she heard her little buddy calling me S....after a month or two (of gentle reminders "you mean mommy?"), She's back to calling me mom (but she also refers to anyone who walks in the door as mom/dad simply because she hears me tell the kids "your daddy/mommy is here!" So to her those are just their names, nothing more)! My daycare kids also know who their real mom is and they jump up so excited to get them at the door, so there is absolutely no confusion there! :-) so feel confident that you will never be forgotten or confused by her titles! You are a great mom in considering your daughter's needs before your own (but also check your emotions and face around your daughter or you may end up with a stressed out people pleaser like me)! :-) Good luck with this all!