A question for the forge guys though, is vintage axe steel known enough about that you could reheat-treat it with actually knowing the exact steel used?
Short answer: yes.
....some people think you can go by color, no matter what the steel, and some think you inform your color-judgment with how hard the steel is to work. You can always test hardness with a file (should skip off, not dig in) and test for over-softening with a common nail (axe edge should be harder).
Let me preface this by repeating that I am only a novice smith at best. It's my understanding (and experience) that vintage carbon steels can be hardened by common means, usually a water or oil quench, and then tempered using the temper colors. But what Porch said is right - the ultimate test for the backyard smith is the file test.
A brief accounting of the process is as follows. Heat the metal to 'critical' heat. For most vintage carbon steels cherry red hot will suffice. But to be exact, 'critical' heat is when the steel loses its magnetic properties. When it's red hot you can test it with a magnet. If the magnet is still attracted then you're not hot enough yet. Some new exotic alloys may require a higher heat to reach critical. At critical heat quench the steel in water or oil. Oil is probably better for thin objects like knife blades but water should do fine for axes.
After quenching test your steel as Porch described. The sharp corner of a file should slide off the metal with little or no digging in. If it isn't hard yet then try quenching again from a higher heat. Try dark orange. If that doesn't work try bright orange. I wouldn't go beyond that. If quenching from bright orange heat won't harden it then you've got something exotic that likely requires a special hardening process.
If a file test shows that the metal is adequately hardened then you're halfway done. Now you need to temper it. Tempering is removing some of the hardness to make the steel tougher and less brittle. The eye especially should be well tempered so it doesn't crack during wedging (I've had this happen).
Tempering can be done by holding it over the forge fire or by heating it with a torch or even putting it in the oven for an hour. The surface of the metal will change colors as the heat increases. The surface needs to be cleanly sanded or filed to see the colors. Apply the heat back away from the edge and let the colors run to the edge. After tempering test the metal with a file again. If it's too hard to file then your need to temper it a little hotter. It should file OK with a sharp file with a bit of effort. That's the correct hardness for an axe. If it files easily then you might have tempered it too much - too hot. If it won't file at all then it needs to be tempered to a little higher temperature.
I've posted these charts before but here they are again. More info on basic hardening and tempering can be found in 'The Complete Modern Blacksmith' by Weygers or in 'The Backyard Blacksmith' by Sims. You can build a simple forge cheaply and easily. Basic hardening and tempering are easier to learn than forging.
Remember, those temperatures are in Celsius.