The ball in baseball, when it is "in play", acts as the decision maker and either has to beat a player to the base or be tagged by a player of the fielding team with the ball or the glove while retaining the ball within it.

The ball is "in play" at anytime the Umpire signals as such and is "not in play" when either the Umpire has called an end of play or pause to allow players to reset to their ready position or the teams are switching from batting to fielding positions. Any time the ball is replaced or not in the fielding teams possession it is "out of play".

A player requiring time will request a time out/pause of play by making a T by placing one hand horizontally over the other which is vertical indicating the shape of the letter. The batter upon returning to the base and placing his foot upon it may have made this signal to the base Umpire who would then refuse or allow the request. If the ball was returned to the pitcher the Umpire would likely allow the player to compose himself, off base for a short few seconds before indicating play is restored.

If the first baseman (the player on the opposing team that is wearing the heavy leather glove) retains the ball the Umpire may not signal a time out due to the ball still being actively "in play". This would note to the runner (batter previously) that the ball is either still in play or the Umpire is denying the request for another reason.

The player didn't ask and assumed the ball was returned to the pitcher as the first baseman went through the motions of returning it while palming in in his glove. The player believing this continued to step off from the safe position while the ball is in play.

In this gif the ball continues to be "in play" and the runner neglected to watch the ball or request a pause of play to relax for a second before play continued and was tagged "out". The player is removed from the field and an "out" is scored against his team. Outs occur through catching a ball when hit before it touches the ground, a fielding player tagging the base the runner must move to before the player arrives at the base if they must run to that base due to being pushed along as the players on his team make reservations of the bases (one player per base and no returns once the next base is reached), or the player is tagger with the ball in the tagging hand. A player can also be called out by swinging and missing 3 pitches while at bat that are determined to have been within a proscribed 3 dimensional box above the "home plate" which is where the batter hits from. This box is roughly the space of the batters knees to elbows and maybe smaller if the batter slouches lower than average, and encompasses the length and width of the "home plate" base.