So if I understand the current situation correctly, it was NOT the daughter relaying information directly from her mom (who was maybe doing the dishes or something like that), but rather the daughter directing a conversation while pretending (through omission) to be her mom. Right? The OP asked to reschedule their meeting, the daughter suggested Saturday, then had to end the conversation because she didn't know what specific time her mom would be free on Saturday. Obviously if her mom was readily available, she could've just asked her what time, and relayed the information back to the OP. But instead she got involved in a scheduling conversation for a third party whose schedule she didn't even know. That to me is quite intrusive and rude, not to mention random and weird.
Also, I think the OP said she sent "her friend" her Saturday schedule, which is what the daughter saw. What if the schedule included information that the friend knew about, but that the OP wasn't comfortable with her daughter knowing? Like, "I can't meet at 3pm because I have my AA meeting/marriage counseling/probation hearing then," or something.
I think it would be fine for the OP to mention this incident to her friend, maybe bring up "the confusion" that ensued when Laura started a scheduling conversation she couldn't finish, and ask if it would be better to just call from now on. Maybe get a feel for whether this was a regular thing or one-time occurrence, and if the mom was totally cool with it or if she was a little exasperated with her daughter. Unless she said she'd chastised her daughter for getting involved and pretending to be her, I think I would stick to phone calls from now on. As others have said, the friend can have whatever rules she wants for her own house, but if the OP isn't comfortable with them, she can use other methods.
Side note: My dad and I share an account for passworded website that includes messaging among members, and it's in his name. Often we both want to answer people who send us messages, which can get kind of confusing. I ALWAYS start my message by typing "(Lynn here)" and then sign my name at the end; and when I'm replying to someone for the first time, I ALWAYS start by saying, "Hi, this is Lynn, Bob's daughter who shares the account." It's more for clarity than anything else, but I also don't want anyone to feel like they don't know who they're "talking" to--it just seems more respectful.