I call myself a single parent. Dd doesn't see her dad at all and Ds goes 3 nights a week. Dd is also 17 so doesn't need as much physical care and supervision.
Just because the other parent see the child, doesn't mean you always have support.
Ds was sent home from school due to covid, on his dad's days. I had to come out of work to get him, because dad couldn't be contacted. I am the one staying at home with him until Tuesday. He has an autism assessment next Tuesday (also his dad's day) and I am taking him to that. His physio appointments, I have to book those on my days too.
When work need me to travel to say, Scotland, if its my days I have to arrange childcare. If its not I still have to make sure his dad knows so that he knows I can't be available.
When lockdown was imposed in march 2020. I was the one working from him and home schooling. Because nothing got done at his dad's.
It used to be 50:50 care. I still did most of the work, because it just doesn't get done. Uniforms, shoes, clothes etc are all ignored by his dad and provided by me. The kids are always with me for Christmases and birthdays. He isn't bothered about seeing them. Ds is slowly getting more and more fed up of being at his dad's so I am following his lead and having him more days. His dad is also self employed and earns next to nothing on paper. So there won't be any financial support.
Ds spends most of his time messaging me and talking to me when he is at his dad's or asking me to log in to his x box at our house so I can play games with him while he is at his dad's. I am in no doubt within the next few years ds will stay here more. O dont want to push him to not see his dad as he does love him. So going at his pace.
The 3 nights I get off a week are usually filled with me doing extra work so I can spend time with ds when he is here (I can work flexibly to a degree). Or doing bigger jobs at home, that are easier when ds isn't here.
Its fairly exhausting trying to co parent with a parent who does the bare minimum.
If ds dad was not involved at all, I would call myself a lone parent.
But when needing support from work during the lockdown, it's easier to say I am a single parent. Rather than the long explanation above. When people are complaining that my child shouldn't have been in the supermarket during lockdown, it's easier to say I am a single parent than 'I am a co parent, whose co is shit and decided he can't see him this week because he broke lockdown to see his girlfriend and her kids. and now they all have a continuous cough.'
The assumption seems to be if the other parents sees their kids, there's loads of support and life is easy. But I don't think my situation is unusual. I think lots of people technically co parent, but still do the vast majority on their own.
When parents aren't together, theres so many variations of how it people are. So if they want to describe themseleves as single or lone or whatever they know their situation better.