First off let me say sorry to hear about your issue. From what you have described, you most likely got an old battery or one that is of very low quality. Yes there is a difference between what an Apple Store or authorized Apple repair center will use to replace your battery versus what a third party may or will use. IMO, it’s not worth the little bit of savings, if any, to not get it done at one of the above places that are authorized to do so. I do know that there is definitely a difference in batteries as to how good they perform under load. That third party place you went to obviously must use some cheap batteries and thus they will pass this along to you in the form of being cheaper possibly. It only about $40 I believe to have it done by Apple or somewhere like Best Buy that is an authorized Apple Repair Center. You will no doubt get a much higher quality battery from one of these places versus the third party shop. Although they may look the same, I can guarantee you they are not the same inside. I have an older 6s plus that I gave to a co-worker and she had the battery replaced by Best Buy, she now has 100% battery capacity and no issues with the phone saying the battery is of low quality or not capable of performing to it fullest. So Yes, get that battery replaced as soon as possible as to not possibly damage your phone. JMHO.