I would submit that both terms are correct when referring to the Army of the United States during the War of Northern Aggression. However there would have to be a division of time when that army existed for the terms to be used correctly.

For example the term "Union Army" would, I feel, be most correctly used from the period of April 1861 until January 1863. The primary purpose of that army existing was "to preserve the Union" therefore it was most correctly the Army FOR the Union.

However, after January 1863 until the end of reconstruction the declared aim of the war for the Federal Government because 'to free the Slaves". This anti-constitutional Stateist change of objectives of that Government became the reason for that Army to exist and to impose that objective on a people who had declared themselves independent of that United States Government.

Therefore the term "Federal Army" would be applicable because it was then carrying out the war aim of that federal Government.