I am right in the middle of a bunch of stuff, so I can't go into detail like I usually would.
Plus something does not sound right about that starter noise, but I'll look at that later.
The suppression they are talking about is for the coil in the solenoid.
Any time you have a magnetic field and a conductor moving in relation to each other, you produce electricity.
When you turn on the solenoid, the coil produces a magnetic field that expands outward to pull in the armature.
When you turn the solenoid off, that magnetic field collapses back down through the coils, plus the armature moves back due to the spring.
This produces a voltage of opposite polarity from the original which can be quite high.
A suppressed solenoid has a diode across the coil reversed according to the normal voltage.
When you turn on the coil, the diode is reverse biased and does not conduct. But, when the reverse voltage starts to develop the diode turns on and shorts it out.
You can look at the diagram of a solenoid and tell if it is suppressed. There will be a diode symbol across the coil.
There are several places that sell starters, winch motors and solenoids and they list which are suppressed.
Or you could add a diode to the one you have.