Confusion about calculating inductance of a coil
My guess is that the 2nd calculation determines the inductance for a
  coil one meter long and I have to divide by the length of the coil I'd
  want to find the inductance of, but I'm not sure.


No, it's got nothing to do with a 1 metre length of solenoid.

The EEweb calculator is definitely problematic because it assumes all the "turns" are stacked on top of each other and not laid out like a solenoid. This can be easily proven. For instance if I set up the loop diameter to be 279 mm and the wire diameter 1mm, I get an inductance of 1 uH. 

If I double the turns I get 4 uH. If I make the turns 100, I get 10 mH - this is exactly how inductance would change if wound on a core with no leakage flux i.e. inductance is proportional to turns squared.



However, 100 turns cannot occupy a close-knit volume of space and at all possibly hope to have 100% coupling from every turn to each other turn. 

Basically, it's a pile of doo doo.