Hello, Mila. Yes, the etymology of sideburns has been repeated consistently in every source I've ever seen. I believe you can take it as reliable.

According to Why You Say It, Burnside also sported a hat so flamboyant that it took his name as well.

I have a vague recollection that my history teacher also related that General Burnside additionally tried a facial hair style which involved not shaving underneath his jaw line all the way down to his collar, but that particular fashion statement didn't catch on the way sideburns did.