I doubt you've ever worked in any type of food service, because there is almost zero chance that the manager that was spoken with made any written record of the conversation. Especially at a place like Raglan Road which is a one-off restaurant (although they used to own another in Vegas) and not a corporate chain with corporate policies and handbooks.
And just showing up to the hospital and saying 'I got hurt at X' isn't any type of proof that you got hurt at X, no.
So yes, if you want to be safe, you should file an actual police or medical report with the proper authorities. Now - since an ambulance was called in this case, it should suffice for what would be required for the insurance, but its always better to be safe than sorry.
But you're now getting into the weeds about what might be required instead of your original question which was simply 'Why would you file a report?'. And lets not mince words - you were asking that question because you were taking the stance of 'this is your fault, why would you try to sue? you're just looking for money', not any type of 'you don't need to file a report in a situation like this' response.