Yes, English is frequently ambiguous, and "sitting" is a good example. "She is sitting on the table" can either mean:
She is in the process of sitting down on the table
She is currently seated on the table.
This has nothing to do with the use of the present continuous. Instead it's a natural ambiguity of the verb "to sit" which has multiple interpretations in other tenses:
"She sat on the table" =
1. She was seated on the table
2. She moved to sit on the table.
(Side note: To be seated on a table means, literally, to be sitting on top of the table. More commonly we sit at the table, to eat a meal, etc.)
You can only determine which is meant from context and common idiom. Two examples:
I told Sally to sit anywhere she liked, so she sat on the table.
This implies the action of sitting down.
Q: Where were the children?
A: There were no more chairs so they sat on the table.
This implies the condition of being seated.
Most of the time, this distinction is not important. However, if it's not obvious, and if it is important to be clear, English speakers will use other conjugations and idiomatic expressions:
She is seated at the table.
The crowd was just sitting down when the accident happened.
There is no ambiguity with the verb "to wear". This always implies a continuous state, and not instantaneous action. Instead, the natural expression for the action is to "put on" an article of clothing:
She is putting on her shoes.
It's raining, so he is putting on boots and an overcoat.
Finally, here is a somewhat-related kids' joke that plays off of this kind of ambiguity:
Q: Why was the scarecrow so admired?
A: Because he was always out standing in his field.