My answer applies only to the UK. 'Bill' is used when you are going to pay for a service immediately after receiving it, e.g. at the end of a meal in a restaurant or at the end of a stay in a hotel. 'Account' is used when you have a longer, established relationship with a business - you get services from them regularly and pay regularly, but not after each time you receive the service. 'Tab' is not used in the UK (as far as I know) and may even not be understood widely. I think these terms will have regional variation.

You can confidently use any of the three and be understood, but there are subtle differences to keep in mind. Use "on my bill" as your safe, general default. Neutral. Use "on my tab" when you're in a bar or a very casual setting, it's more bar/pub specific. Informal/Casual. Use "charge to my account" when dealing with hotels or established business relationships. Formal.