Hi Giacomo - welcome to WattFlyer.
Nowadays, there's a great range of small helicopters on offer. As you said, the choice can be confusing.
Generally speaking, 2-3 channel helis are little more than toys, which can be fun but ultimately, disappointing because of their limited capabilities.
Thumbs up, for wanting something better - and coming to the right place for advice.
Choice of model depends what your goals are? - whether you are hoping to become proficient in flying a more advanced machine, or just want to have fun, flying a cool toy that has better handling characteristics?...
With your <$50 budget, you might struggle to find anything outside the 'toy' category. Looking on the HobbyLobby website (for example), the only helis under $50 are coaxial and 3 channel.
Contra-rotating (co-axial) helis don't need a sideways tail rotor, because the opposing torque of two blade sets is cancelled out.
They have various methods of controlling direction: Some turn (yaw) by varying the speed between the 2 sets of blades - others use a sideways tail motor.
Forward/back motion can be achieved by either tilting a set of blades in that direction, or by using a vertical tail motor to tilt (pitch) nose up/down.
The difference between 3 and 4 channel coaxial, is that with a 4th channel, you'll be able to move sideways.
With coaxial models, stability is good but manoeuverability is limited, so a single-rotor design would be a better long-term choice.
Ironically, coaxial models which by their nature, look silly with 2 sets of blades, are often available with better-looking 'scale' bodies, which is rarely the case with single-rotor models.
A single rotor heli needs a sideways tail rotor, to cancel the torque of main blades. It will also have directional servos, which tilt the main blades in the direction you want to fly (cyclic).
This will require 4 channels: throttle/pitch/roll/yaw. If you are already experienced flying 3-channel coaxial designs, then a small 4-channel single rotor should be a relatively easy upgrade.
If you are likely to crash a lot, then aim for one which offers good spares support. If something breaks, you'll be able to mend it cheaply.
Also, better to choose a smaller model at first, which will be lighter and hopefully more crash-resistant.
My Blade mSR is tiny, stable, fixed-pitch heli - great for flying around the house - very manoeuverable indoors - also capable of taking a few knocks!...
It bounces off walls, ceiling, furniture, (cat - lol) with almost no damage to house or model. I've never broken anything on it but I did wear out a set of blade grips which cost about £3 to replace.
If you are upgrading from a 3 channel, this class of micro heli would be a cheap and fun way to go.
My original mSR isnt so good at flying outdoors - requires a windless day, or it gets blown around.
The later version (mSR-x) might be better at outdoor flying because of the AS3X stabiliser - but I havent tried it myself.
One thing to remember with model helicopters, is that the bigger they are, the more dangerous they can be - treat them with respect and try to get some advice locally, before you try to fly a larger machine.
I could list lots of models but I won't - firstly, you'll probably find more by googling them yourself (there are plenty out there), plus I'm in the UK, so prices/suppliers here will be different.
Whatever you buy, good luck with your flying !