What is the best budget GPS watch for running?

Within the Garmin range, at the budget end for running would be the Vivoactive 4 or the Forerunner 245 (the 245 adds some more advanced running metrics).

The 745 adds multisport/triathlon capabilities.

The 945 adds mapping and a larger battery life.

The Fenix 6 series offers largely the same capabilities as the 945, but in a nicer case, with some larger and smaller size options.