[DrDoom]

I had initially considered that the Union Grounds occupied the shaded area of the above map (exactly 550ft sq) and maybe it was lumped in with "public parks" because it was likely used for events other than baseball.

However, I stumbled across this article from a german Cincinnati news paper:

1868 Westliche Blätter

Translation:
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This magnificent pond, designed for the comfort of skaters, is located in the northwest end of the city, just behind Lincoln Park. You take the road, railroad out of Freeman Street to Lincoln Park, walk through it and get straight to Union Pond. This pond is at least three times as large as the so-called "Skating Rink," opposite at the Lincoln Park, and has the advantage that one is in free pleasure and can romp around much faster and more undisturbed. Meeting rooms are furnished and heated for gentlemen and ladies, and a pleasant club, with a gallery in front of it, affords a magnificent view not only over the Teib[valley?], but also over the hills to the west and north of the Sratt[City?]. On clear, sunny days there is no more beautiful abode than the neighbourhood of this pond, with the merrily frolicking runners frolicking around on it. — On the days when the pond is dedicated to the 50 cents. Only members or ticket holders have access to the Brivattagen[?]. The public, however, and especially the dear youth, compensates for this by skating on the ponds of Lincoln Park, where it costs nothing.
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The article clearly describes walking through a very typical park (Lincoln Park) to get to an admission only skating rink referred to as Union Pond.

The BR Bullpen reference above mentions the Union Grounds being used as a skating rink in the wintertime. I think it is safe to say that the "Union Pond" mentioned in the Westliche Blätter news article is referring to the Union Grounds flooded as a skating rink.