Possible fraud when selling a car privately?

Hello everyone, I'm a bit confused right now and hope someone can tell me more about this.

On Saturday, I sold my car privately because my partner and I don't need two cars at the moment. The contract was recorded in writing.

A few hours after the car was sold and the buyer drove home, I received a message from her saying that the car had broken down and was now being towed by the ADAC. A group transport; she and her boyfriend will have to wait until next week.
Naturally I asked what had happened. She then said the injector was defective and that her boyfriend had apparently read this from the error log. The ADAC later confirmed this. But I told her that I was a bit confused and shocked because her boyfriend had also read the error log before they bought the car and it only stated that the central locking of the rear door was defective, which I had made clear from the start. He also did a test drive and that symptoms of a defective injector would have been noticed. Things like that don't just break down like that. Above all, I saw the car after the sale at the supermarket in our town and they said they wanted to spend some time there. Maybe that doesn't matter at all, maybe not.

I recently had the glow plugs and relays replaced at my trusted garage, after which my partner and I were told that the car was in top condition again. I haven't had any problems since, and I'm extremely sensitive to car symptoms. She refused to even look at any of my garage's invoices for the glow plugs and relays, the new clutch with dual mass flywheel, or the brakes, even though I practically shoved them in her face. She also didn't look at the papers for the MOT and AU, or the car's deregistration.

Anyway, I would like to see some kind of proof, receipts, or documents from her. But they're just talking their way out of it or suddenly twisting the facts. Like, for example, that she suddenly doesn't have to wait until next week, but can pick up the car tomorrow. I was supposed to pay her for the injector, or rather half of it. After I asked for receipts/invoices for the part, it suddenly wasn't so important anymore, and I was supposed to pay her at least half the fuel costs. She didn't get a receipt for that because she "paid cash"… (that's a very new, strange reason to me 🤔).
So it wasn't about yesterday's fuel costs, but rather the towing costs. She'll have to invest at least €200 for that.

Doesn't the ADAC keep a record of the incident, or shouldn't you have received some kind of proof? Likewise, they don't have any photos or anything similar.
I find all this very strange.

At some point, even my goodness runs out… I'd already bought her a new battery beforehand, and I even went below my pain threshold with the price of the car, even though I was assured they wouldn't negotiate on the spot, which they did. I put brand-new Toyo rims on her expensive new wheels (for an extra charge, of course), but even there, I went to my pain threshold. And in retrospect, I even went below it, because I agreed to negotiate.
I feel a bit cheated and disappointed, even in myself.

Of course, something really could have happened, and I'm very sorry about that, as it was my car before, and I loved it very much. But I just have the feeling that either they're only telling half the truth here, or they're trying to squeeze out more money after the fact because the negotiations were so "easy" anyway…

I've written to my workshop again to see what they have to say and will probably get an answer on Monday.

Thank you in advance for the answers and help 🙏