A quick, and possibly somewhat inaccurate history may help... (Also note that the measurements for barrel thickness are under the handguard. They might not be exact if you measure with calipers at other points along the barrel, but it's close.)
The original barrel on the M16 and AR15 had a diameter of something like 0.625 inches. This barrel worked just fine as a rifle barrel. It might lose some accuracy when it got hot, but it did it's job while being nice and lightweight.
Then someone came up with the idea of mounting a 40mm grenade launcher (M203) under the barrel of an M16. They designed the grenade launcher to secure itself to the 0.625 inch barrel. Each infantry fire team (about 5 men) got one of the grenade launchers. So most rifles did not have the launchers.
Then the Marine Corps started working on improving the M16 which ultimately resulted in the M16A2. They wanted a heavier and more accurate rifle and increased the barrel diameter to 0.750... Or they would have, but that made the rifle really heavy. So they just kept the heavy 0.750 inch thickness up front and then thinned it down back to the old diameter under the handguards. This made the rifle somewhat heavier while not really improving the accuracy at all. But it did make the rifle less likely to bend when used as a crowbar. (Sadly, I am not making that up. It was one of the original justifications for the design.)
And then the military decided to make the M16 shorter. They reduced the barrel length to 14.5 inches and called it the M4 carbine. They also kept the 0.750 diameter barrel up front and the thinner diameter under the handguards. The problem here was that now the M203 couldn't fit on the barrel. So they cut out the barrel to 0.625 in one spot so the grenade launcher could be attached.
And some M4 carbines (M4A1) now have the 0.750 barrel thickness all the way to the chamber, minus an extra cutout under the handguards. (not sure what it is for right off)
Today, in the civilian world...
There is the original barrel thickness of 0.625 which is commonly called either a "lightweight" or "pencil" barrel. This would be my choice for a carry rifle or general use rifle.
There is a full .750 profile which goes by different names. This would be the choice if you are trying to make a really accurate rifle and weight isn't much of an issue.
There is the "Government" profile which will be .750 in front of and at the gas block and about 0.625 behind the gas block.
And there is the M4 profile. Because the M4 has a 14.5 inch barrel and this would make an AR an NFA item, most of these barrels are lengthened to 16 inches. This makes the barrel look a big wrong IMO. They are .750 back to the gas block with the M203 cutout, and then .625 behind the gas block.
There are also various custom offerings.
I own a 16" M4 profile rifle that's a fine shooter, the barrels do work and there's nothing wrong with them. But I would much prefer something else because the design really is pointless for civilian use.
The government profile is just fine and I would go with the mid length government profile if I were you. But, if a midlength pencil barrel were offered I would take that instead.
Hope that helps.