The difference between Stop and Stop and Proceed aspects is the absence (Stop) or presence (Stop and Proceed) of a number plate on the signal mast. It is not possible for a signal to display both - the number plate is either there or it isn't, permanently. Stop signals are used at home signal locations, usually to protect interlockings. Stop and Proceed signals are used for automatic block protection in ABS 251 or 261 territory.
In general -
The function of the signal system is designed by the operating rules people, as the signals have to function in conformance with the operating rules. The operating department works with the signal engineers to ensure the mechanical/electrical/computer operation of the systems meets the needs of the designed operation. All of it must meet the FRA's regulatory requirements and adequate safety considerations, but beyond that it is up to the individual RRs to design their systems, so one RR may use a particular feature or not.
The ability of an interlocking to display a Restricting signal is a matter of its wiring/relay/interlock design or (with computerized interlockings) programming and is specific to the particular interlocking and part of the RR's signal system design. Some RRs might not want a restricting signal for a following signal so the system would be designed like BostonUrbEx described. If you're a commuter RR with no cab signals, passing a signal that required Restricted Speed operation would mean a train would crawl to the next signal, causing delays. If you're a freight RR running long heavy freights in heavily graded territory at lower speeds, the savings in preventing stop/starts would make a case for following on Restricting signals.
The ability of interlockings to display restricting aspects into an occupied block was originally referred to in many places as a Engine Return Signal, as it allowed a Leverman/Operator/Dispatcher to put locomotives back on their trains when dropping or picking up cars without verbal or flag communication. It is also a method of reducing starts/stops as much as possible. IIRC, in a tower with an electric interlocking machine where the handle is turned down to 'force' a restricting, the interlock is overridden and the signal is pinned low so that it cannot upgrade to anything beyond Restricting. It may or may not have maintained a route/switch lock when forced, depending on design, and the rules required crews to ascertain that their route was properly lined in the field. For a Restricting to be displayed with a normal 'handle-up' operation would depend on what type of track was lined to and the operating method in force on that track. Tracks like running tracks or yard tracks would not warrant a better signal indication if the rules required Restricted Speed on them. Except for places where 'forced' Restricting signals are used, there is no dispatcher/operator control over what signal aspects an interlocked/controlled signal displays. They only tell the interlocking the route and 'go' or 'stop'. Stop is the normal condition, and fleeting is only a 'repeat go' command for the interlocking to display the best aspect possible given the track/occupancy/route conditions.
What the #*** did we just hit, Over ???