But why do they record crime incorrectly? If they were underfunded, being honest about the amount of crime in the UK would be a good case for giving them more money, right?

There may be other reasons, but trying to look like you’re doing a good job can be a conflict, as while higher crime rates might justify higher budgets in the future, they’d also lower the rate of solving and preventing crimes which are seen as metrics of police success.

Other than relying on 100% of police to be 100% ethical (and even without congnitive biases), which obviously isn’t ever going to be the case, third parties are probably always going to be necessary to improve accuracy.

Define "crime". Is all reported crime to be recorded as such? Police cannot take the word of everyone who reports, and often well-meaning people report things that are not in fact crimes. Police do not investigate all crimes witnessed or reported by the public. No force in the world has such resources. But they cannot honestly record alleged crimes as actual crimes, nor can we hold them responsible for counting all the crime that the public doesn't report to them. So it is normal for public surveys to be different than those from police reports.

I don't want to define anything, I was just going off what the grandparent said about the police crime reports being less reliable than surveys.