Yes, that's correct. The relay is triggered by the same brown wire that triggers the starter solenoid.
Here's a diagram of the relay connector and the solenoid connector.
View attachment 128755
You'll need a helper to bump the starter and see if you get power to the BROWN wire at the relay. If you have power there when you bump the starter, replace the relay and check the center terminal that's connected to the RED/VIOLET wire at the solenoid connector the same way you checked at the relay connector by bumping the starter. If you're getting power to the relay and to the solenoid, you have a bad ground or a bad solenoid.