I'm far from a mitre saw expert, but I was considernig the Axy one you mentioned.
I've a couple of thoughts:
The design is very similar to the old Elektra Bekum KGS300 (certain components are copies or identical), which is what I ended up with secondhand. Mine (the EB) is quiet and pretty accurate.
The blade is direct drive from the motor shaft. This has a drawback -- as an induction motor is physically bigger than a brushed motor of similar power it gets in the way a bit. I find it limits the size of stock more than I'd like. Wide boards are no problem, but thickness is. The other issue is that induction motors can't be cheaply made variable speed, so it's a single speed saw. I'm not sure how important that is - it doesn't bother me.
The good points of induction motors are the noise (obviously), that stalling the motor briefly won't damage it (not a good idea with a brushed motor!), and that there are no 'wear-out' parts on the motor itself. As long as you don't abuse the bearings, and care for the slides, the saw should stay accurate for a long time.
I like the trigger arrangements on the EB (very similar to the Axy one). My grip isn't what it was, and the DeWalt/Elu arrangement, with a giant trigger doing the blade guard and a 'pistol' type of motor trigger, is a dangerous nuisance - you tend to grip with your first two fingers, meaning that there is a risk of accidentally starting the saw when you're setting up the cut. This is not a problem with the horizontal handle. Otherwise the 701/707 series have a longstanding good reputation, spares are readily available and they're worth a look secondhand (I think they're overpriced new, nowadays).
I don't think the Axy can trench, and doesn't have a laser. I've no idea if these things are useful or gimmics - they're of no interest to me. It also is conventional design and overhangs at the back. Again, that doesn't bother me at the moment, but might when my EB gets built into a bench.
I ended up with the EB one after just missing an eBay auction for the Axy one. A nearly new one went for £70(!), so there are bargains around.
HTH,
E.