Headspace Gauge Comparisons is a start to understanding the differences between all the gauges.
In actual use, pull out your bolt, remove the ejector.
Hook the gauge under the extractor and slide it into the chamber.
The only real gauge you need to own is a Field gauge and that's so that you know when your barrel/bolt combination is worn past safe limits.
There is little concern about the chamber being too tight.
If it's too tight, it will not chamber the ammo.
This will be readily apearant and in no-way unsafe.

If you want to own all the gauges, then you can check with a Go gauge first.
It SHOULD fit into the chamber and the bolt will lock closed.
Then check with a No-Go, it SHOULD NOT fit into the chamber, but it might and the barrel is still within specs if it does happen to fit.
Lastly, and once a year for sure, check with a field gauge, it SHOULD NOT EVER fit.
If the field gauge fits, then try a new bolt.
If a new bolt and a field gauge still fits, get a new barrel.

You can also check out my 223 vs 5.56 comparison and take special note of the actual differences between the various flavors of reamer.
You can also see the difference is mostly in the headspace length and in the throat diameter.

The biggest tip-off to a 5.56 chamber is to look at the throat and note if the groove diameter is fully cleaned out.
If it is, there's a good likelyhood you have a 5.56 barrel.
This is a really subjective thing to look for and you really have to know what to look for.

Quoted: If you know where I might find them cheaper, or have any specific recommendations (brands?) please let me know.

www.brownells.com
These are all forster gauges and it's generally recommended to buy all your gauges from the same maker so that their master gauging and QC standards are similar.
About any gauge coming in a clear plastic tube with a yellow cap that presses inside the tube is going to be a Forester gauge, no matter what it says on the side.
319-418-033 1.4736" Colt 5.56 Headspace Gauge $16.21
319-223-470 1.470" .223 Remington, Field   $24.50
319-223-467 1.467" .223 Remington, NO-GO   $24.50
319-223-464 1.464" .223 Remington, GO   $24.50

Quoted: or if you have them, use head space gauges...

HS gages only tell you the length of the chamber not if it's a 5.56 or .223.

even in barrels that are stamped one or the other the chambers may be the other. the only way to be certain is to cast the chamber and check the throat against the print.

short throats in Armalite barrels

Headspace

.223 V 5.56

As you can see, with the exception of the GO gage, the military uses longer dimensions than the commonly available civilian gages.

Quoted: Since all those NATO reamers differs a little bit from another, which one of them is "real" NATO reamer,

I really don't know.
What I SUSPECT is that the PTG is on the minimum end of NATO specs while the Clymer is on the maximum end of the specs.
I don't know what NATO throat angle is called out.
A lot of this is really up to the reamer makers to look at the specs and then decide how they want to grind the reamer.
It's not uncommon at all to move throat lengths and angles around.
Most people are not aware of the minor nuances of throating, but that's where the accuracy lives and dies in a barrel.

Quoted: Is this true?

depends on who is telling the story

when I worked there we stamped them .223 Rem. until the stamp wore out then ordered a 5.56 stamp because stamps with fewer letters are cheaper, we never changed reamers and the reamer was a .223 on the min side of SAAMI.

now they say they're reaming standard barrels 5,56 and UM barrels .223 but back then they used to say that the two barrels were cut with different reamers and that was BS too.

If you're concerned cast the chamber and look at the throat, that's the difference.

doesn't matter either way as long as the HS is long enough.

even if it is a .223 it won't matter esp if the HS is deep enough. backing the projo off the throat by deepening the HS achieves the same end as opening up the throat. it's exceedingly tight match chambers used with 5.56 that can be problematic.

Quoted: I´d really appreciate, if you could add JGS .223 Wylde reamer specs to your comparison chart.

My reamer comparison now contains the 223 wylde to compare to others.

the NATO spec for chambers isn't "all across the board" even if the domestic reamer makers are. the concern is one of HS, mil chambers tend to long and loose and the ammo follows that. civ chambers can be anything and a large hot round in short tight chamber can be troublesome.