The police misled you if they said that you broke it therefore you're guilty. There's more for the prosecution to prove than that! The offence is that you broke it intending to damage it OR being reckless as to whether it would be damaged.To be reckless (as meant here, under the Crimial Damage Act) is to be aware that a risk of damage exists and nonetheless unreasonably take that risk. Pulling a door hard, if the risk of damage was not apparent to you, you were not reckless.Likewise, even if you saw some risk of damage but it was normal or necessary for someone to pull it hard then you were acting reasonably in taking the risk, so that's not reckless either. For example, if the door was stuck you'd quite possibly see a risk of damage in pulling on it to get out but the law doesn't expect you to be kept inside the building yelling for help !It would say it's reasonable to pull it as hard as is necessary to escape.
It could be said that by offering to pay you were admitting guilt but that argument doesn't hold up. Insurance companies pay out on pure accident without anyone being criminally or civilly at fault ! You might, if you were a lawyer, offer to pay 'without admission of liability' .Anyone might quite reasonably offer to settle to avoid the expense and trouble of litigation or any other inconvenience.