I struggle with this as a parent - my son no longer sees his step mother since she was pretty unpleasant to him a couple of years ago. He just sees his father on their own.
The point is not about double standards. Of course there are two standards, one of you is their parent, the other is not.
Being a parent carries with it a huge backstory of love, of devotion, of pain and most of all a huge attachment to the child, going both ways. This means that any comments or tellings off are within a context.
If you are not the parent, you do not have this same context. Just as I feel it's alright for me to lose it a bit with my children sometimes, maybe swear or shout, within reason of course, but would not wish one of their teachers to do so - I wouldn't want a step parent to do this either. Because they are not connected to the child in the same way and do not carry the burden of this connection, which means that even when you shout, your child knows you still love them and will be there for them and do anything for them.
It's a very basic difference. and imo anyone who is not the child's parent ought probably to approach the child in a very guarded manner in terms of how angry they get.
OP I don't think you said anything that awful, and given your position has been over some time, it probably was an overreaction, but really the only thing you need to remember is you're not their Mum, and therefore, you can't behave like their mum. However much childcare you actually do.
It's a very difficult line to tread. You're not a parent, but you're being given parent duties. I sympathise.