Starter or Solenoid

Before I go through the work of replacing the starter I want to make sure I've got this right.

When I hit the ignition I can hear the click click of the solenoid pushing out the gear to engage the flywheel but it doesn't turn.

I am presuming this is the starter?

Starter tested out but I swapped it out anyway. Lo and behold it also fixed the warm starting issue.

I love when fixing one problem also fixes—unknowingly—another. Good news!