I am a "lone" parent - meaning I have my children 100% of the time with no involvement or financial support whatsoever from the children's father.
There are challenges that come with this that I don't believe others necessarily understand - the main example being that you are 100% responsible for your children 100% of the time with no safety net. All decisions are yours and yours alone. If something goes wrong, you are alone to solve it. No one is there to help you make important decisions, you can't bounce off the other parent and decide together. It is absolutely relentless with never any time off - even if you ship the kids to the grandparents for a night you are still on and thinking about what needs doing, and it puts me in mind of when you first have a baby. You know in your head it will be hard and think you understand before the baby arrives, but until you have that baby you don't understand really what that means (or in my case I didn't anyway).
However, that does not necessarily mean I have it worse than a "single" parent. My best friend is divorced from her kids dad, who was abusive to her. He is still in their lives and she has all the stresses to navigate that come with that. She has often said she doesn't know how I cope totally alone, but I can't believe she has to deal with her abuser in the ways she does.
Both are hard and have challenges, but both are very different. However, although I can intellectually understand that what she deals with must be so hard, I don't really understand what it is like day in and day out as I have never been in that position.
I think that is why some make the distinction - the two situations are so different and come with such different challenges. It can make people fell more heard if they get advice from someone who has actually been in their shoes and has more of an understanding of the unique pressures they are facing.
But ultimately everyone's circumstances are different and there is a scale - a lone parent may be high earning and have great family support whereas there may be a low earning single mother dealing with an abusive ex who barely has the kids and pays minimal child support. There is usually someone somewhere worse off than you. And while I can't get too excited about what someone calls themselves, there is a big difference between a lone parent (as I understand the term) and a single one with shared responsibility for the children.