No, you don't need it on because you're already hearing the AxeFX through your monitors. That Input Monitoring allows you to hear what's being input into that track. If you play with it ON you'll get that nasty out of phase sound, turn it OFF and it goes away.
It sounds like the problem you're having is somehow that nasty phase sound is getting recorded and the only way I can think that's occurring is you're sending the output from the AxeFX, into Logic then back of out Logic and into the AxeFX again and back into Logic. If it's ending up on the bounce, I'm not sure how else it could happen aside from routing outputs/inputs in Logic.
You want this-
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Not this-
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But I need to see your mixer to see how you have the Inputs/Outputs routed (the Logic mixer)
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Or better yet, can you post the file you're sending to the engineer?