If you are getting no response at all from the starter when you turn the key your primary suspects are going to be:
I know you already tried another battery, but I would probably want to try hooking up a charger for an hour or so just to be sure there is no issue there.
Otherwise you will need to have a shop check for power signal at the starter (on the small wire) when the key is turned. If there is 12V signal there when the key is turned, the battery is assumed good, and the cables/terminals are all tight, then that would only leave the starter to be at fault.
If however there is not a 12V signal when the key is turned, then we woudl have an issue with the control circuit... meaning ignition switch, neutral safety switch, and wiring in between them and the starter. You didn't mention the year/model, but it is worth noting that Mitsubishis specifically from mid 90's through late 2000's were very common to have the ignition switches fail, resulting in the interior amenities all working fine, but when you go to turn the key to Start there is no response.