The riddle of steel

5160 is a popular choice, as well as 1045, 1055 and 1075. T10 Tool steel is also a reportedly tough steel.

It's a super tough question because everyone has their favourite and people always argue for what they like most. Another big issue is the differences between steel types make some better suited than others depending on the construction, shape, use - that sort of thing. Yet another point is the argument that the smith is the most important factor; that is to say it's better to have someone who knows how to properly treat the steel, rather than have a preferred steel that is mishandled. Personally I fall into the "it's the smith, not the steel" that counts faction.

All that said, typically people seem to like spring steels like 5160 or 9260 over the 10xx series. But then again, you can spring temper all kinds of steels, and I like me some 10xx steels plenty (some real nice hamon action in them is one reason). Another couple of steels that get that "super steel" thing mentioned about them are S7, L6 bainite.

There's tons more to consider of course but this is start.

Facing Crom I'd say:
"Not Stainless, more than 0,4 % carbon and a good heat treatment.
If that's not enough, then to hell with you!".

I'm satisfied the most with my swords made of 6150 and 5160. Are they better steels than others? I don't know, the thing is probably they were heat treated by people who really know what they are doing... (Albion, Del Tin, Hanwei Tinker...) Oh, and what Viktor Berbekucz is using (51 CRV4 ). :D

You are right: steel = iron + carbon (0,4 - ca. 2 %; more makes cast iron).
You can add some alloys to make steel a bit more flexible (silicon for spring steel) or give more hardenabilty (tool steel) with the right heat treatment.
With ca. 12 % crome it becomes stainless but usually to brittle for swords.
"Carbon steel" means usually "not stainless steel", even though stainless steel has carbon too.
Similar problem with "spring steel", which usually means steel with some Si, but you can heat treat steel with only carbon to be springy.
Don't overrate alloys, for a sword the most important thing is how it's formed and heat treated.
So 99 % is the smith, 1 % is the steel.
As Luka said, it's more important that a smith is skilled with "his" steel, than what steel this is.

The usual "carbon steel" term for blades is simply an abbreviated form of "steel with enough carbon to harden during quenching, but not alloyed." The 1045-95 series is your basic set of carbon steels as is often understood in the blade community.

And as everyone said, preference is king (and skillful heat treatment is queen). You can keep it simple with the 10xx series. You can go for cheap toughness like 5160, or expensive toughness like cpm3v. You can go for aesthetically pleasing steel like pattern-welded nickel steel (15n20) with high carbon steel (1095). You can even go green or oldschool with reused/recycled steel like leafsprings or coils or bandsaw blades or old files.

My personal take is of course, it depends. I want a really pretty hamon? W2 steel. A federschwert for HEMA sparring? 5160 or 9260 and their equivalent. A beater? I'd be happy with 1060. A custom piece for the collection? Pattern-weld. High performance steel for a custom sword? Whatever John Lundemo wants to use.

Simplistic: From 1045 - 1095, mostly 1060 - 1075, without alloys but with more inpurities.
To spot a fake is not easy, chemical analysis?
Common sword steels are not so expensive that I fear fakes from well known makers/companies.
If you buy an expensive overhyped steel like "tamagahane", "L6" or "damascus" from no name seller, it can be 1045 or stainless.
But you can make a good sword even with 1045.
A stainless fake could be dangerous.

I would love to buy a genuine medieval sword someday but it would be way to risky for a beginner. But I don't want to get off the subject. Where does the carbon come from when making swords?

Am I correct in assuming it's coal today?