They're fancy goldfish, but I've no idea what kinds I'm afraid, not my type of fish. Trying to think of any members who are into fancy goldfish who might know the type! But they all can get pretty large, have heavy bioloads (eat, pee and poop a lot) so need a very large tank and lots of filtration to keep them healthy and prevent stunting.
Goldfish are normally coldwater fish but can also live in warm water. the more fancy varieties actually do better in water that is between 15-25C, and they struggle when it gets really cold. The two fish you have are fancy varieties that do best at temperatures above 15C.
The white dots in the eyes can be from poor water quality or cataracts. The moors are pretty badly inbred and have a number of genetic issues including going blind at a young age.
You should try to avoid putting rocks or ornaments in the tank and just use live plants. That way if the fish swims into them it won't injure itself. And try not to rearrange the tank because the fish will learn the set up and where everything is, but if you move stuff around it will have trouble working its way around the tank.
Do big (75%) water changes and gravel clean the substrate every week.
Make sure any new water is free of chlorine/ chloramine before it's added to the tank.
Monitor the ammonia, nitrite and nitrate levels in the water and do a 75% water change any day you have an ammonia or nitrite reading above 0ppm, or a nitrate reading above 20ppm.
Keep feeding down and make sure they have plenty of plant matter in their diet. Duckweed is a small floating plant that most goldfish love to eat.
Don't feed them too much dry food because they ingest air at the same time and can float around the tank. Frozen and live foods are better for them.
Types and Sizes of Common Goldfish Breeds First of all, never underestimate the size of fully grown goldfish. There are no 'dwarf' breeds of goldfish, whatever 'the man in the shop' might say; even the smaller 'fancy' types have the capacity to grow to a good six inches/15 cms in length...
Those are what are called Fancy Goldfish . Fancy Goldfish fanciers have names for all the various kinds. Those might be a Black Moor and a Telescope Eye as has been mentioned. Fancy Goldfish are a whole unique specialty of the fish hobby and have devoted fans as much as do Bettas and Guppies.