Dan, the Otari MX-5050BQII and MX-5050BQIII are exactly what I said they were. They are 4-track, 4-channel machines capable of recording on all four tracks at once. They have four inputs, four outputs, four VU meters, four rec/rep amplifiers, and 4-track 4-channel record and reproduce heads. They are not quarter-track machines at all, but I suppose they could be used to play four-track stereo tapes by selecting tracks 1&3.
Since you brought it up, Otari did make a quarter-track machine as well but only in the MX-5050BII series. That machine was the MX-5050BII-4. It had the usual 4-track, 2-channel record and reproduce heads for four-track stereo tapes plus an additional 2-track reproduce head.
Other models were: MX-5050BII-F (full-track mono record & reproduce plus an additional 2-track repro), MX-5050BII-2 (2-track record & reproduce plus an additional 4-track, 2-channel repro), and MX-5050BII-2E (same as MX-5050BII-2 but with DIN 2-track).
The MX-5050BIII series discontinued the 4-track, 2 channel heads, so the only other models were MX-5050BIII-F, MX-5050BIII-2, and MX-5050BIII-2E. None of these had the additional repro head.
All these were vertically oriented machines for 1/4" tape. Otari's MX-5050MkIII and MX-5050MkIV were the corresponding horizontal machines. In that version, four-track machines were always for 1/2" tape.