If this happened immediately after replacing the starter, I would say the wires on the solonoid are connected wrong. It could be the switch but I think it is the wiring.
Bad ignition switch or starter solenoid
no
The starter solenoid on a Honda Element is integral to the starter. It cannot be replaced without replacing the starter as well.
The starter solenoid is located on top of the starter. It is the small cylinder attached to the starter. It is a part of the starter which can be replaced/repaired independent from the starter itself.
Starter solenoid is sticking. Replace it.
solenoid broken
that test indicates that you need to replace the starter motor.if no clicking in this test , you would replace solenoid first. but clicking indicates solenoid works,so replace starter motor
Good power all the way to the starter, 12 volts at solenoid/start circuit wire when key engaged, but no click or starter engage
I DO BELIEVE THE SOLENOID IS MOUNTED ON THE STARTER AND YOU WOULD HAVE TO REPLACE THE STARTER TO REPLACE THE SOLENOID remove the starter and take it down to your local auto parts store. they can bench test the starter and give you the proper solenoid if it needs to be replaced.
The starter solenoid can be replaced by removing the wiring harness. Remove the retaining nuts. The solenoid will come off. Reverse the process to install the new solenoid.
may be on top of the starter and replaced with same. Follow the + battery cable it will connect to the solenoid
The wire from the battery to solenoid is corroded & should be replaced.