Bill, thanks for finding a clearer Sears Parts Direct diagram. The model number to the saw you found is prefixed with 315, which indicates a Ryobi manufactured RAS, but interestingly enough, the basic design is so similar to the 113 prefix manufactured by Emerson, one wonders how they could get away with that.
Anyway, the part #10 you identified in that 315 model saw is similar to Part # 42 in Figure 3 that @TomCT2
pointed out, and is the same as the pin I described in Post #9 earlier as being drawn out of scale relative to the motor. The Sears Part Number for that indexing pin is 63645.
As it turns out, it was not drawn out of scale, it really is a big honkin pin, much larger than the mystery pins that dropped out of the saw, as @JayArr
predicted, and that big indexing pin was soundly in place on my saw. And keep in mind, as @woodnthings
pointed out, I was operating the saw in a 45 degree bevel (that I previously described as "tilt") and the motor carrier remained rock solid in the 45 degree bevel detent. I later disassebled the handle, front cover, motor mounting screws, cover plate, etc... exposing the bevel pin that was still there. Much bigger pin than what dropped out of my saw. So not the culprit.
also pointed out call out part #7 in Figure 3 of the 113.199250 diagram. That "Pin Assembly, Index" is actually a spherical plastic knob attached to short lever arm that is attached to turned pin with multiple diameters... a decidedly complex and 3 dimensional "assembly" of permanently put together components, none of which resemble the pins that dropped out of the saw.
Based on Bill's, Jay's, and Tom's advise... and especially based on what Jay explained about the epoxy cracking off and needing to fabricate an alignment jig to reassemble the motor, and @woodnthings
"DO NOT take apart the motor... Trust me on this.
"... I will trust you. I'm not going to take apart the motor.
But I do want to solve the mystery of these pins... and haven't even come close yet.
... You mentioned a second guess of "assembly pins used in conjunction with a machined bolt.
" What do you mean?