It's not a matter of disrespect as much as of disobeyance. If there isn't (or shouldn't be) any subordination, disobeyance may not imply disrespect. The problem arises when people differ about whether there's subordination.
I disagree. Being rude is about being offensive; this is different from someone being offended.
For example, I was mugged at gunpoint once. The person very politely asked for my walled (it turns out when you have a gun pointed at someone you can afford to be polite). When I asked if I could keep my wallet and just hand over the cash, he responded (again surprisingly politely) in the negative.
If I had just ignored his request, would that have been "pretty rude"? [Ignoring whether or not this course of action would be advisable.]
And for a second hypothetical example. If I asked you to respond to me agreeing with everything I said, would it be "pretty rude" for you to ignore it?
In both cases the answer has to be no. The nature and intent of the request have to be considered when determining whether or not ignoring it is rude. And in this context (recording a medical event), they almost certainly want you to not record in case they do something wrong and the recording is used to prove it. It's a self serving request and it potentially works against the best interest of the patient in events where they believe their care is not sufficient.
You can’t be rude ignoring a request that is both dumb and wholly a product of bureaucracy. Such requests are themselves rude, in a kind of weird impersonal way.
Of course you can. For instance, if you would reply to the request by insulting the physical appearance of the one making the request, that would be rude.