I have tried to explain this before, at least the way I understand it... the station type is defined by its use. This is why you never ask cops for advice. You need to consult lawyers and judges.
The rule very specifically says that a base station is a station at a fixed location that communicates directly with mobile stations and other base stations. Therefore, the act of using a station to communicate through a repeater changes the 'station type' to a Control station. A Control station is a station at a fixed location that communicates with mobile stations and other control stations through repeater stations, and may also be used to control the operation of repeater stations.
So, using a radio at your home to communicate through a repeater is not a rule violation.
The only defined hardware type is a Mobile radio, with handheld radios being a sub-type, with the definitions as follows:
Mobile station. A station, intended to be used while in motion or during halts at unspecified locations, that communicates directly with base stations and other mobile stations, and with control stations and other mobile stations through repeater stations.
Hand-held portable unit. A physically small mobile station that can be operated while being held in the operator's hand.
All other references are to station types, which are defined by how they are used... for example:
Base station. A station at a fixed location that communicates directly with mobile stations and other base stations.
Control station. A station at a fixed location that communicates with mobile stations and other control stations through repeater stations, and may also be used to control the operation of repeater stations.
Fixed station. A station at a fixed location that directly communicates with other fixed stations only.
Personal Radio Services station. Any transmitter, with or without an incorporated antenna or receiver, which is certified by the FCC to be operated in one or more of the Personal Radio Services.
Repeater station. A station in a fixed location used to extend the communications range of mobile stations, hand-held portable units and control stations by receiving their signals on one channel (the input channel) and simultaneously retransmitting these signals on another channel (the output channel), typically with higher transmitting power from a favorable antenna location (typically high above the surrounding terrain).
You will notice that even a repeater is defined by the job it does, rather than physical characteristics. That is because you can easily build a repeater with 2 mobile radios, as an example.