Where do I connect the In-Wall Relay Module Switch?

The manual only specs this for a “resistive load.” That means an incandescent light or a halogen light, and specifically does not mean LED or CFL or fluorescent (which are inductive loads).

From the manual:

Resistive Load:
800W for 110VAC

From the seller:

So @anon36505037 is correct, this particular device should not be used with your fluorescent tubes.

@Navat604 or one of the other electrical experts in the community may be able to say more.

But in general you need to look for one which is specifically spec’d for fluorescent tubes. The physics are significantly different. As one electrician explains:

Load types will differ in inrush current and arching.
While a cold resistive heating load will have a high inrush, once the temperature stabilizes it will have a relatively low interrupting current when the contactor attempts to open the load.
An inductive load will hit the contactor at both ends: There’s a high initial inrush, and there’s also a significant amount of energy that has to be dealt with when the circuit opens and the magnetic field collapses, so there’s a good sized arc when turning the load off, too.
Just did a troubleshooting call for exactly this problem on an electromagnet: They were burning through manual disconnects that were in theory rated to handle the current, but they weren’t designed to open an inductive coil that size and it was caused serious contact erosion.