I think single parent is fine. It’s a household with one single parent in it.
I have been a single parent and a lone parent.
’Co-parent’ doesn’t work because couples also Co-parent.
Quite frankly my life is easier without the EX involved. I don’t agree with the narrative that single parents with the ex involved always have it easier.
Exh had ds 3 nights a week. Ds became anxious about school because exh would drop him late. Uniform, trainers, clothes, toys etc would go missing at my exhs house all the time and he wouldn’t replace them, so it was up to me. There was always problems with his latest girlfriends that upset both my kids. My eldest stopped seeing him early on after he tried to tell her she had to treat his new girlfriends children like brothers and sisters, 6 months after he had split with another woman whose kids were to be considered her brothers and sisters.
He caused the kids all sorts of emotional problems. I had to pay for counselling. There was no Child support payment as he was self employed and his income is officially low.
I still had to pay all my own bills, run the house, organise the kids (even if things were happening on his time), sort Christmas presents, birthday party presents etc but trying to do it with someone who is determined to make it more difficult is soul destroying. The fact that I often got 3 nights a week on my own, mostly which were at the weekend, meant that I spent those 3 days working then catching up on everything I could to make life easier when ds came home.
Since his involvement has reduced to an occasional couple of hours visit, it’s been loads easier. Dd went to uni, with the only input from him being a box of food that was mainly beans. There’s been no dramas, no issues and so much easier that he wasn’t involved. Me and dd have sorted it all between us.
The term lone parent isn’t always accurate. I know people who have no involvement from their ex, but loads of family support. Weekly sleep overs for several nights. Babysitting when they need, free child care, help with the bill. In some cases the lone parents, parents take the place of the absent parent.
The point I am making is that all situations where you parent alone, have situations where it’s easier or harder. Some co parent with and ex and it’s harder than doing it alone. Some do it without their ex but have it fairly easy because of their family. Some people who are entirely alone, find that easier.
Squabbling over what people call themselves because you view them as not having a hard enough life, isn’t helpful to anyone and no one label will encapsulate their own situation, exactly.