ESA and Cold Weather Payments

Quick Question I received a CWP today first in years as I now live on coast and it's a few degrees warmer than where I was previously but given the recent cold weather etc. I am on PIP Enhanced Rate Mobility and Standard Rate Care as well as ESA ( IR) SG.
I understood from when I used to be on DLA middle rate care that eligibility for CWP was being on Income Support and in receipt of Severe Disability Premium. I had to give up the Severe Disability Premium last year as my eldest daughter turned 18 so I was no longer eligible. Phoned ESA today but they said anyone in certain groups including ESA SG or WRAG is entitled. However, this contradicts the Direct Gov. site which clearly says eligiblity is with ESA AND a severe or enhanced disability premium.Any ides? Or does the Disability Income Guarantee class as eligibility? Not sure the woman I spoke to was " benefit trained " ( Yes, that old chestnut) so if anyone can shed some light on this as I don't want to be in the overpayments nonsense further down the line

Thanks Gordon Poor form for the Gov.uk page to be carrying erroneous information though....Thanks again