All of the facts previously stated are correct as far as I know, but
I am now going to step into it here with all the definitions that got tossed about through the years for pro, semi-pro and amateur formats. You can play a 4 channel tape on a 4 track recorder - that is one like a Teac 3340 for example, that records and plays four tracks in one direction with the aforementioned 4 track head(s). 4 track was typically used as more of a pro term for the bigger machines to do multi-track recording. 4 channel or quadraphonic for the quad (Q4) tapes you mention and their home playback machines. 1/4 track for the standard stereo interlaced 2 channel consumer format that was just mentioned in the first reply, but confusingly the tapes are typically called 4 track because, well, they have 4 tracks - an interlaced stereo pair running either direction. Another usage of 4 track was to quadruple time mono record a tape using the 2 track either direction for 4 track mono with 4 different programs.
Prerecorded reel to reel tapes were first introduced in the 2 channel (stereo) 2 track format - a few early ones even with staggered head configuration where the left and right channels were displaced on the tape by a set amount for a few of the first consumer stereo machines that used 2 mono 1/2 track heads next to one another. Again, there is some confusion between 1/2 track mono (consumer) one mono track each direction on each side of the tape, and 2 track stereo which is typically the pro mastering standard at higher speeds (like the Tape Project tapes)
I did not know if there were any other matrix encoded or carrier encoded quadraphonic systems for reel to reel tapes as there were for vinyl. I just knew about the 1/4 inch 4 channel ones (Q4) and the Q8 variation for four channel "8 track" with 2 four channel programs as opposed to the normal 4 two channel programs on 1/4" tape.
BTW, if you do play a Q4 quadraphonic reel on a 1/4 track machine, you will hear 2 channels of the quad mix (I forget if it would be front left and right or front left and rear left)