I am known as

All are correct except the first two. You could say for no. 1, "I am known as a hard working person and for being able to work independently."

For number 2 you could say, "I am known as hard working and being able to work independently."

I do not agree.

Of the four sentences, 3 and 4 are acceptable, and 3 is better than 4. However, I would say

I am known as a hard working person, able to work independently.
I am known as a hard working person who is able to work independently.

For your first sentence, you need something else, either "and I am able to work independently," or, "who is able to work independently." Just "able to work independently does not work here.

I'm not sure what else you are disagreeing with, as I said 1 and 2 aren't correct as he has them, and that 3 and 4 are, which is what you are saying.

sentence 2

uktous Hi,
Question:
Which sentence is correct?
Sentence1:
I am known as a hard working person and able to work independently.
Sentence2:
I am known as hard working