Cooler bottle storage

Hi everyone! FTM here, my baby is less than a week old and my milk just came in a few days ago, and I am currently EBF. I didn’t realize how much would leak on the other breast during nursing sessions so I’ve been collecting it in a haakaa instead of wasting it in a nursing pad but I’ve been finding I need a better solution so we don’t have to walk it to the fridge every time in the middle of the night. I’ve heard a little about the pitcher method for storage, and I’ve seen some people online using a cooler water bottle at their bedside that they then bring to store properly in the morning. My question is, is a breastmilk cooler bottle any different than a regular cooler bottle? I see them advertised and I already have a lot of keep cold bottles I feel like would work if sterilized and what not first. Does anyone have any insight? What do you do with yours? TIA! ❤️