IMO
Gym Plant - Gym opens new location in completely new region as a stand alone operation. Typically staffed with people from the home gym to ensure the same processes, policies and programs are carried thru in a way that is consistent with the home gym.
Gym Merger - Two programs mutually come together to share athletes and resources. One gym always is the majority stake holder although that is rarely known to the outside world. The benefit in a merger is keeping as many clients on board during the transition period to make it worthwhile. Eventually one gym fades away into the background so to speak but it takes years. In some cases it never does and one gym may end up breaking the merger to go back on their own again.
Gym Acquisition - One gym basically takes completely over for the second gym, providing capital, staffing assistance, training, etc. Nothing remains of the old program is it is generally wiped away. This can sometimes be mutual and sometimes hostile.
Satellite Gym - Gym opens a location with a primary purpose of providing lower level training opportunities and to serve as a feeder gym for its home or main gym. The function is primarily feed the home gym with the best of the best. Also gives a gym a name in the community with the hope of one day being on a team at the home gym.
Franchise - Gym "sells" the right to use their name, logo, colors, to another gym. Very little is done in the way of day to day operations. Coaching staff remains essentially the same as it was prior to the franchising. Only change is the name. Big benefit to the franchising gym owner who is able to generate more income via rebates, a wider net for top level athletes, publicity, etc. Franchise gym benefits from instant name brand recognition, access to better and more expensive choreography, music, etc. In some of these cases the franchising gym may rarely if ever set foot in the gym they franchised.
Not a perfect definition mind you. Just a quick thought.