Based on the clues provided in the text (including the full article linked above) and with the use of the 1836 topographical map linked here, I think these are correct identifications of the ponds where skating and hockey took place:
Note that 4th Avenue is now known as Park Avenue.
The bodies of water, of course, are long since gone... likely filled in not too long after the publication of this article. With the construction of Central Park right around the time of the article, and Hamilton Square being parceled out for development very soon thereafter, the physical landscape in that area would have changed very quickly. I'd imagine any bodies of water would have been filled within the decade, other than those which were reserved for the park itself. It was actually hard to find a map which represented the pre-development landscape including bodies of water, hence the older map above.
Today, the Arsenal Ponds are the location of some of the most desirable real estate in the world, a block of the Upper East Side facing onto 5th Ave and overlooking Central Park. The Hamilton Square Pond, where hockey was played, is at roughly the location of present-day Lenox Hill Hospital. Lenox Hill is the descendant of the German Hospital, which first started leasing land near the pond about 5 years after this article was published. Fittingly, Lenox Hill was the first hospital in the US to have a physical therapy department, as well as the first to have a center for the study of sports medicine.