ESA Support Group - Income related & Contributions

I'd like to be clear on what I am receiving and wonder what you think: I am in receipt of ESA which I believe is income related. I am also in the support group. I've not worked for over 10 years.
My council are saying that I am in receipt of contributions related ESA & have consequently increased my council tax following a recent change of address.
A recent letter from the DWP seems to state that I am in both groups but it really isn't clear.

Hi Gordon, no this isn't a new claim. I was on Incapacity Benefit over 10 years ago and was transferred to ESA about 2 years ago. I receive approx £110 a week ESA.

I am apparently receiving income related also but at "zero" rate. The DWP letter states this. I don't understand it but the letter says this. The council said in their letter that the reduction in housing benefit (council tax increase) was because I moved house, but by email they say it is because I was moved from income to contribution ESA and the DWP informed them of this. My income hasn't increased so I can't understand why my council tax has gone up.

I have a private pension of 190 pounds a month. Could that be a reason they are doing this? Although I had that previously.

What question/s should I put to them? They just avoid responsibility.