I can imagine what reference headphones are supposed to sound like. The K-612s sound great in the mids and highs but the low end is missing. There is absolutely no way that anyone could mix with these because you'd be working in the dark if you can't hear the bass properly.
Open back high impedence studio references types are typically supposed to be driven from desktop amplifiers- in a studio setting there's not really much demand for portability. It's also worth noting that just because an output source has the theoretical power to drive headphones doesn't mean they will do so well. A headphone amp "designed" for 10-150 ohm cans usually starts running out of puff around half of that.
I use them with the Fiio E10K and I can only get them to a comfortable listening level by maxing out the volume.
That's usually a symptom of the amp not having the grunt to effectively drive them. I've had similar experiences trying to run my AKGs and Sennheisers from portable sources without an external amplifier. The E10K is really designed for headphones in the 16-150 ohm range so 120 nominal from the 612s is right at the top end of what they can drive, let alone drive well. I expect your experience might be different driving from a dedicated desktop headphone amp. And I don't mean a USB type one either but a proper externally powered one- my USB Audioengine D1 headphone amp is designed for 10-300 Ohm and makes a pretty decent hash of 150+ Ohm cans, but pales in comparison to driving them from my dad's ridiculous Burson Soloist. Then again, he's spent much of his retirement building a borderline absurd hi-fi setup.
The in-ear Sony's sound more balanced actually.
They're better matched to the E10K. In fact the E10K is ridiculously overkill for driving them.
I'd suggest trying either a bigger amp or lower impedance reference headphones. K550 IIs are 36 ohm and even the ridiculous K712 Pros are 62. Albeit at twice the cost of 612s, probably cheaper than buying a headphone amp with a 300 ohm range.
What music do you listen to?
Pretty much everything, from heavy metal and punk to classical, jazz, D&B/breakcore/techno, the odd bit of hip-hop (typically stuff at the noisier and denser end of the spectrum like Dälek or Anti-Pop Consortium), ambient, film scores...
I can't imagine the cheaper AKGs sounding good no matter what genre of music you listen to which makes them terrible for mixing.
Which headphones are you referring to here? Most of the £100-300 price range AKG stuff is amongst the best you can buy in that price bracket. Everything I've tried, from the Y50s through my everyday K550s up to the 712Pros has been consistently excellent.
Are you particularly looking for open backs? If so, the have a look at the Grado range. If you aren't fussed and can't push the budget, try Samson Z45 or Z55s.