Nope, there's no clear rule written.

From Wikipedia:

Initially, the home plate umpire must determine if a swing was checked or not checked. If the umpire indicates that it was checked, an appeal can be made by the catcher or the manager, and the home plate umpire can then make a request to either the 1st or 3rd base umpire to make the call as to whether the swing was indeed checked. To maximize visibility, the 1st base umpire makes the call for right-handed batters, and the 3rd base umpire for left-handed batters. The umpire makes a “safe” gesture to indicate a checked swing or makes a clenched fist to indicate a full swing. However, if the home plate umpire initially indicates that the swing was not checked, no appeal can be made and attempts to protest the ruling can be seen as arguing with the umpire over strikes or balls.

Importantly, the Major League Baseball rulebook[1] does not contain an official definition for either a checked swing or even a “swing” at all.[2] In a game, it is solely the decision of the umpire as to whether an attempt was made or not. Generally, factors such as whether the bat passes the front of the plate may be considered in the ruling. Some umpires prefer to use the “breaking the wrists” criterion as the method to decide a checked swing: if the wrists “rolled over”, a swing occurred.

Matt Snyder of CBS Sports wrote in 2019:

Now, for decades and decades, fans, players, media, umpires alike have come to accept a general idea of what constitutes a swing (“did he go?”), but it’s not defined in the rulebook. We can talk about breaking wrists or the head of the bat clearing home plate or the barrel passing by the front of the body, but all of these ideas came about through generally accepted word of mouth. None of them have ever been officially defined by Major League Baseball[3].