So the nature of the particles that you are colliding, their speed, and trajectories, doesn't vary based on what the community want to try

Only the nature and their energy (that's better than speed) of the particles define the machine.

The rest (signals and trajectories) is dictated by quantum field theories. Which quantum field theory to choose tells you which signal to expect. You take the Standard Model as your background, and then your favourite theory as a signal. Then, we do the computations and design strategies about how to filter all collisions to make sure that the selection focuses on the signatures of the considered "signal theory" by keeping the "Standard Model background" small.

Is it clearer?