Your dialog is a funny take on a more serious feminist discussion during the 1970's and 80's. Back then men would often refer to their wives as "housewife", which implied that she didn't have a job and she just stayed at home all day ie: "she's a wife who stays in the house". This wasn't true of course because even though these women weren't paid they still did an enormous amount of work taking care of the homes, feeding their families and raising children. In order to force an acknowledgement of this work into the conversation, women began using the phrase "stay at home mom". Because almost everyone agrees that being a 'mom' is a full time job (even if they don't get paid for it). So that is the difference. "Housewife" literally means a wife who stays in the house, and it doesn't acknowledge the possibility that she may be working twice as hard as the husband. "Stay at home mom" literally means a woman is staying at home and doing all the work of a mother, so it is more respectful. So with that in mind you can see how your dialog is a kinda funny. When you switch the genders it becomes a little more obvious (to men) that "househusband" is a dismissive term. I remember one of my male friends would get upset if people said he was "babysitting" his kids. This was in the 1990's. Someone would ask if he wanted to go out somewhere for a drink or whatever, and he would say "I can't tonight, sorry, I'm with my kids". Sometimes people would say "oh you're babysitting, ok no problem", and he would get quite mad. "I'm not babysitting, I'm parenting!". He was right of course. The phrase "babysitting" implies that he's just temporarily looking after a child until the mother comes back. I can see how a father would find this a little bit insulting. It's interesting how our chosen words reflect our inner selves. It shows us the things we subconsciously value, and the things we subconsciously dismiss as unimportant.

A stay-at-home dad, of course, is a dad. A househusband needn't have kids.

Thank you so much for the answer! :)

As a native English speaker I've never heard the term "househusband" be used in a serious context. To me it sounds like a mocking term in that it's trying to emphasize that he's a man in a role traditionally done by a "housewife". The corresponding term "housewife" definitely was (and sometimes is still) used, though it sounds old fashioned and somewhat offensive today depending on your region and dialect. In my opinion, it's best to stick with stay-at-home mom/dad or just use stay-at-home parent for both.