There's really no "right" or "wrong" way - whatever sounds best to you is the right way.  And that's going to depend on your gear and your ears.  The way you describe your rig is NOT the way Line 6 designed their product to be used, but that doesn't mean you're damaging your equipment or that you shouldn't believe your ears.

The difference between the power amp output modes and studio/direct is that studio/direct uses mic simulation whereas the power amp modes don't.  All these modes use speaker/cab simulation, but "speaker/cab simulation" is a deceptive term.  A real speaker dramatically alters the frequency response of the signal - clearly rolling off a lot of the high end.  The "speaker/cab simulation" is not meant to do that, only to much more mildly EQ the tone so that whatever real speaker you are using sounds more like the speaker/cab simulation you select on the Pod.  The Pod relies on your speaker to roll off the high end.  The mic simulation is what really rolls off the high end, making the tone sound like a traditional guitar speaker when the output is run into a full-range speaker.

What kind of speaker is in your amp?  I would suspect it isn't rolling off the high end as hard as other speakers.  You wouldn't notice it with your amp, because your amp is voiced to take the speaker voicing into account.  But the Pod isn't.  Also, certain amp power sections are brighter than others.  A lot of people like tube amps because of the affect they have on frequency response, seeming to diminish the harsher frequencies and boosting the warm ones, in addition to their dynamics/response changes to the tone.  Your power amp might be a little bright, but again, your amp's pre-amp is voiced to match, whereas the Pod isn't.

I have a Line 6 Spider Valve Mk I with a Celestion Vintage 30 speaker.  I plug into the power amp input as well.  I use Studio/Direct output mode, but I select "no cab" as my cab.  With "no cab", there is no cab or mic simulation; and the power amp output modes sound the same as Studio/Direct mode.  Using a cab+mic in Studio/Direct mode through my amp makes it sound too deep and scooped - it sounds fake or something.

My friend used to have a Fender combo amp (I think it was a Hot Rot Deluxe) - he ran a Boss pedalboard into it, and it sounded great when he ran the Boss pedalboard with speaker simulation on, but it sounded horrible when he had it turned off.