What to call a partner's child?

A friend of mine posted on Facebook something along the lines of:

"Why is there no word for the child of a long-term partner? They're not technically your 'stepchild' if you're not married. 'My partner's child' feels too impersonal when you're talking about somoeone you live with and love. I don't want to call them my child because I didn't carry them and birth them and raise them - someone else did, and it feels disrespectful to their own mother to do so (I know it would hurt me if someone else called my child theirs).'

Does anyone on here know if any words to refer to this kind of relationship? Or has anyone in a similar situation made up a word that works for them? (In this instance, they were specifically talking about their partner's daughter, but I'd be interested to hear about sons, too).