Starter will not crank over the engine?

Hello, I recent purchased a used vehicle. I noticed at times my car would not crank. It will switch to the on position, but once turned to A/C car does nothing. Battery is okay, and just had starter replaced as I thought it was the issue. Twice now the car has not started. First time I went ahead with old trick of putting car in neutral then cranking. Second time I kept key turned to crank it (for what I felt was longer than a second or two), jiggled key a little further and car started up. I noticed two occurrences before changing starter that I resolved the issue "magically." Once was out of frustration and I banged all over the steering wheel and dashboard, near ignition switch. Ta-da car started. Second was the same as the most recent time except, again, it was more forceful; I jiggled the key around while cranking key and *vroom* again it started in enough time for me to get to work. I noticed a recall for vehicle:
NHTSA RECALL NUMBER: 10V-364
MFR CAMPAIGN ID: R44
CAMPAIGN DESCRIPTION: IGNITION SWITCH RECALL
RECALL DATE: 08/04/2010
Is entire ignition switch faulty? Can it just be the pins? Is it possibly a loose wire elsewhere in the process a car goes through to start? If I take it to a dealership to inspect the recalled parts and it fails the tests mentioned in photo attached will it resolve the issue? Further information is greatly appreciated. Thank you.