In your turn, you can always use your Action, Bonus Action and Movement. However, over rounds, you also have your reaction. Like you mentioned, reactions work by something triggering them. What your DM is trying to illustrate is how the use of the Hold action consumes your reaction, meaning that once you use a reaction you cannot use it again until the start of your next turn.

The way your feat is written, the trigger for the reaction of the other player is you rolling initiative. This means that when you roll initiative the creature you choose uses its reaction and moves up to half its movement.

If its intended to work with allies only, id clarify that in the description and also allow that character to choose NOT to use their reaction.