Yes, it's a loan word for Italian, as many others. Μπαλκόνι, νέτος, νούμερο, ντελίριο, ομπρέλα, πακέτο, πατέντα etc
Pool in the meaning of body of water in nature or puddle is λίμνη and λιμνούλα respectively. The pool in the sense of the swimming pool is always πισίνα. That's because Romans brought them here, there were no swimming pools in Greece before the Roman occupation I think... ;)
Κολυμβητήριο is not the same as a pool. Κολυμβητήριο is an establishment with swimming pool(s) in it for people to go. For example, you can't say "Μην πηγαίνεις στο βαθύ μέρος του κολυμβητηρίου". Think of Κολυμβητήριο as a gym with swimming pools instead of exercise machines.
Thanks troll1995 - got it. Although we do also use "swimming pool" in English to refer to the establishment/building, and not only the body of water.