I never tried to record a headphone output with a special wire...Headphone out ---special wire---to preamp
But why not...
Maybe you'll realise that its the proximity of headphones on your head that create that extra touch you try to find on speakers
Good luck
... or if the recorded tone still sounds like ass it'll indicate that the issue lies with the interface, it's settings or your DAW.
Not possible you've got a trim plugin or similar set to mono instantiated on the default amped-guitar track?
How about possible bleed from another mixer channel into the amped-guitar track's bus?