Disclaimer: A lot of what I say in this answer is already said in Thyerus's answer. But I think it gives enough information to be more than a comment. Maybe a community answer would be a good idea to answer this question.
So, a bit of context here: I'm a male in my late 20s living in the south of France.
I think there are four parameters to take into account to know when you're expected to do "La bise". Namely:
- Social link with the person
So, let take them one by one.
Most of the time, women are more expected than men to do "la bise". For instance, when I come to work in the morning, I do "la bise" to all my female coworker and shake hands with all the males one. This is of course dependant of the 3 other points, but note than in a situation where one will expect a man to do "la bise", a woman will be even more expected to do so.
Thyerus said in his answer:
Not sure if it's specific to Alsace but here, I've only witnessed two men doing "la bise" in a father-son situation.
I wouldn't say it's specific to Alsace, but it's really wrong in Provence. A rule of thumb is that in the south, people tend to do la bise a lot more easily than in the north. I don't really know where to draw the line, but I live near Marseille, and I do "la bise" to all my friends, no matter male of female. Once again, there are 2 more points to take into account, but in south of France, the Gender part is (almost) invalidated.
From my experience, younger people tend to do "la bise" more often. Moreover, the greater the age difference, the lesser I'll go for "la bise". I think you can see it the same way as "tutoiement" vs "vouvoiement". If I'm meeting a 60+y/o, I'll probably "vouvoie" them and shake their hand. If they're 20, I'll probably give "la bise" and "tutoie" them.
Your relation with the other person is also very important. I'll separate it in four categories: Friends, Family, acquaintance, and workplace.
From what I get of your question, the problem occured in family. Once again, this may be cultural, but I always do "la bise" to all my parents friends. Moreover, "la bise" is often see as a more affective and non-formal way to greet someone. If they're your mother friends (as in your anecdote), they probably see you (depending on your age) as "the cute little daughter of my friend" more than "the [whatever job you do] of [whatever company you work for]". So it make sense for them to go for "la bise"
I think friends and family aren't really a problem. You know them well, see them often, and know which one expect you to do "la bise" or not. The 2 main problems are acquaintance (friend of a friend, or friend of family types for instances) and cowokers.
For acquaintances, I think you can use the 3 other points: I get from your question that you're a girl/woman. So, that's 1 point for "la bise". If you happen to live/meet them in south of France, and if they're around your age, there is a 90+% chances they'll go for "la bise". If they're much younger/older than you and in Paris or Brest, they'll probably go for a handshake.
The most troublesome part (for me) is the workplace. As I said in the previous points, I do "la bise" to all my friends, male of female, but I shake hands with all my male coworkers. I also said I do "la bise" to my female coworkers, but I shake hands with my female managers. And doing "la bise" to a manager, no matter if male or female, old or young, in Marseille or in Lille would seems really akward. And just as I'm writting this, I realise that some male managers do "la bise" to female employees, but I've never seen the opposite. To be honest, it's probably one of the most akward part whenever I change team: getting to know if I should do "la bise" or shake hands to my new females cowokers.
Sadly, I think there is no way to fully answer your question about "when should one do it?". Has you said it's a lot of greyzones. I tried to reduce it to a spectrum with (at least) four dimensions. I hope it helped you a bit, but sadly it still isn't perfect.
Same age, same social status, at least one is female, south of France -> "la bise"
Big age gap, one is the manager of the other, both are male, north of France -> don't even think about "la bise"
Anything in the middle -> Greyzone!