With LED brightness and different resistors, do I talk about the current or the voltage?
How do I talk about what the resistor is doing?


In this case you call the resistor a current limiting resistor, because it's function is limiting current to the diode. 


  Do I say that the red LED needs less current than the green LED to
  glow as brightly, so it has a stronger resistor to reduce the current?


There is no way to compare a red led with a green led, there are several factors at work when you look at an LED including:

1) Human eyes assign diff the brightness of different frequencies differently, here is the response of a human eye, we see green really good. 

Source: NPL

2) The LED's have different efficiency curves and have different voltage drops and have different intensities. Red is more luminous than green, but to the eye green has an easier time showing up.
 
Source: LED magazine


  Or do I say that the red LED needs less voltage than the green LED to
  glow as brightly, and the resistor soaks up voltage because of V=IR?


You could say, the red led needs less voltage for the same amount of current because its voltage drop is lower than a green led. But voltage drop and current have little to do with how bright an led is. 


Source: LEDnique