English is not my first language, but the first sentence (I accept my fault) doesn't sound very natural to me. I have looked up the word accept in several dictionaries and haven't found any similar example. It is often used with responsibility or blame. Another word that seems to convey the same meaning is admit, but that sounds a little formal and serious. What is the most natural way of saying that you are responsible for something wrong? Personally, the second sentence (I accept it is/was my fault/mistake.) sounds more natural to me.
"I accept my fault" implies that you are admitting that you were at fault at a given point.
As defined in Oxford English Dictionary:
An unattractive or unsatisfactory feature, especially in a piece of work or in a person's character.
"I accept it was my fault" indicates that a specific result was produced because of your mistake. It should also be noticed that, here, you might be trying to say that the said result was not caused by someone else's error and that you were responsible. The previous statement signifies characteristic while this one hints responsibility.
Another definition of the word "fault" in Oxford English Dictionary:
Responsibility for an accident or misfortune.