@Eeksteek I really hate competitive misery, and I completely agree it isn’t fair to put neat labels on people, but conflating actual lone parenting with single parenting of any level really invalidates lone parents.
That's the thing, you can be a single parent without being a lone parent. They're different challenges, but even 3 nights a week (which most certainly don't have) still doesn't save you from knowing that you alone are responsible for the rent, the dishwasher, your pension.
Otherwise it absolutely does become competitive misery. You mention being a lone parent but to one child.
What about someone who has 3 children, and has eow 'off', but spends it feeling sick with worry about what will happen if he loses his temper again? Or who has a 'co parent' who does 3 nights a week, and is paying his CMS, but it's £7 a week because he works sporadically and cash in hand - and uses the rich weeks to treat the kids to Legoland, and never to school shoes? Or one who immediately shacks up with their mistress and soon has 2 new DCs and a stepmum who says there's no space so the older kids now spend their night/s on blow up mattresses in the living room?
In the competitive misery stakes, these 'single parents' and their children are not necessarily better off than lone parents of one or two children without medical issues and with family support, or enough money to buy help.
Generally speaking, parenting in even a two parent family is too much pressure compared with traditional larger family structures. Doing it outside of a couple is another level of hard, and division only plays into the hands of people dying to talk about "welfare queens" and the deserving poor, instead of acknowledging and supporting families and children in tough circumstances.