12.6V is nominal for a 12V battery @ 2.1V per cell. You could clean the terminals with a bit of sandpaper, If your terminals look a bit green/white It means the battery has been venting Sulfurous Acid and causes the effect of corrosion "from the old battery" will find a pic of what I mean and a really nice fix.
Back later.

Right here we go:
Spoiler: Battery
Battery corrosion


Ok this a bad one but if you have any form of this starting it indicates possible battery problems, now you notice this is a maintenance free version, so this still happens .
Now the nice and easy cleanup procedure I use.
First take car outside on driveway or over a sandy patch. Boil some water place in jug and add 2 table spoons of Bicarb Soda and pour this over your battery, wait a while for fizzing to slow/stop use garden hose or bucket of water and rinse Battery and that part of engine bay, Clean terminal post and inside of lug.
Done.