I usually don't care what strangers think, because they don't know me and I can't just assume that they'll know better, and I am sure I mess up with strangers too.
People I know can be different, because if I tell someone I know something, if they care about me at all, they should respect my wishes. If they don't, then they're disrespecting me, and I think people should care when they're being disrespected by someone they know. If I come out as trans to people I know, and they continue to call me "she" and by my birth name, I am going to care. At that point, I'd have to decide who to continue being friends with, and who not to. Having a friend who just tells me not to care isn't being very helpful, and doesn't realize the disrespect others would be giving me.