Under FCC Part 90 the term "Repeater" is not used. The Station Class "FB2" is a "Mobile Relay Station".
The definition in 90.7 is: Mobile relay station. A base station in the mobile service
authorized to retransmit automatically on a mobile service frequency
communications which originate on the transmitting frequency of the
mobile station.
If the retransmission is not done "automatically" the station is not necessarily licensed as "FB2", it could be "FB" if the the retransmission is strictly under the control of the dispatcher.
The RadioReference database does not strictly follow the FCC station class and defines "R" as "Repeater". This is a case of popular usage of a term not necessarily the same as the legal definition.
While a station licensed and operating as "FB2" would be "R" in the database, I think it is questionable how a station that retransmits the mobiles only under dispatcher control should be classified.
If I understand the FDNY system correctly, it is never operated in a strictly automatic mode. The term "mixer" implies that the dispatcher activates the transmitter and may or may not elect to use the audio mixer capability to retransmit the mobile audio while the transmitter is keyed.