The original Union Club was composed of high school pupils who organized under that name in 1859 with Asa W. Smith, president; Robert Niggeman, vice-president; J.P. Freeman, secretary; E.F. Finney, treasurer. In its first campaign it defeated a nine called the Lone Stars (who had only an ephemeral existence) in two match games and another club called the Excelsiors one game, and during the holidays of ’60 it defeated the Empire Club by a score of 15 to 14. On New Year’s day 1860 the Empire Club retaliated by an exact reversal of the score.
-E.H. Tobias, TSN, November 2, 1895
So let's look at this reference about the Union Club. It comes from the Herodotus of St. Louis baseball history - Edmund Tobias. Tobias' history of early St. Louis baseball is probably the most important source we have for the era. It is invaluable and Tobias is a wonderful writer. Not only a great chronicler of the game, he was also someone who played during this era. He was playing with the Commercials during the Civil War and with the Empires during the post-war years. He was living in St. Louis by 1860, at the latest. He lived through and participated in the events that he wrote about. Tobias is a fantastic source.
Here we have him stating that the Union Club was founded in 1859, which is the earliest date he applies to the origins of a St. Louis baseball club and which, if true, would make them the first baseball club in the history of St. Louis. Now Tobias never comes out and states which club was founded first. He makes one general statement on the subject:
Among the very first of regularly formed clubs in St. Louis were the Empire, Union, Cyclone, Commercial, Resolute, Morning Star, Hope and Olympic, each of which was honored by a junior attachment, from the ranks of which the parent club was recruited. Each club had its first and second nines, matches between these being of frequent occurrence.
-Tobias, TSN, October 26, 1895
Base Ball Match--The following is the result of a match game of base ball played yesterday, on Gamble's Addition, by the Empire and Union Base Ball Clubs of this city: [Unions 15, Empires 14.]
-Missouri Republican, December 20, 1861
One of those charming contests for athletic superiority came off yesterday afternoon, upon the grounds of the "Empire," between the Union and Lone Star clubs. The day was exceedingly fine, and the play drew a large number of spectators, who appeared to enjoy the sport almost as much as those directly engaged in it. The match was contested with much spirit, increasing to the close, when victory perched on the banner of the Union...
-St. Louis Daily Bulletin, October 9, 1860
So I would argue that not only were the Unions not the first club in St. Louis baseball history but I think it's very likely that among the pioneer clubs, the Unions formed the latest. While the Cyclones, Morning Stars, Empires, and Commercials were all playing during the summer of 1860, the Union Club most likely didn't form until school started back that year.
So the question of who the first St. Louis club was is a complicated one and it's possible to cherry-pick your sources to support one club or another. Or it's possible to find one source and see it as definitive. But the cumulative source material is complicated and contradictory. If you didn't know that the winter of 1860/1861 was one of the coldest on record in St. Louis, you might be inclined to believe that the Unions were playing baseball in December of 1860. But how many people know what the weather was like in December of 1860? There is a lot of stuff that goes into this and it's important to realize that there is not one definitive source. There is, as of yet, no silver bullet that solves all of our problems and gives us all of the answers we want. You have to consider the cumulative weight of all of the sources and shift through it. And even then, you'll just be able to, at best, make an educated guess as to the first club in St. Louis baseball history.