@ProbablyRomanticised I think that the ideas surrounding rigid thinking are surprisingly...rigid, actually. You are right, it is not all about routine.
My mum is completely chaotic, but she shows rigid thinking in other ways. One day, she decided that she was going to surprise me with a cat, seemingly out of nowhere. No plan or anything whatsoever, I was having a surprise pet and she was going to get a cat. My stepfather then 'ruined' the surprise by telling me we were having a cat, and my mum was distraught. Because it wasn't about the cat for her, but about the surprise, she came home with a dog.
In one way she's very impulsive l, but in another there is this set menu situation where once she has decided something in her head, it has to be that way and if it isn't that way everything is ruined and she will have a (sometimes violent) meltdown or try and twist it into fitting her original plan. When I was diagnosed I was initially under the impression that I fit none of the 'rigid thinking' stuff because I was cool and spontaneous, but I realised that this was very specific to situations I had set up and could control.
So for example I can pre-arrange to not do anything on Thursday and if a friend calls and wants to hang out, no problem. However, if I have planned to watch x TV show in my head, and someone asks 'oh, can you just do the dishes whilst you are here', I might have a complete breakdown. After all, that's not what I was going to do and now I'm filled with anxiety and panic because I lost control. I find this incredibly annoying, but I haven't been able to change it. I can go to a restaurant I've never been to and order food I've never tried before because I'm prepared for that experience, but don't tell me when I'm at home that we are changing the menu from pizza to chips, because I won't cope.