a rafter is simply a sloped beam in a roof much a like a joist is a horizontal beam in a floor. the primary force distribution in any beam (rafter or joist) is bending.
A ridge board is simply a way for a stick framed roof to connect the two slopes of rafters at the ridge - its a convenience but in no way requrired as you indicate with historical examples where no ridge board is present and yet two rafters are connected together at the peak. By the way, if you have no ridge member, then it is clear that all forces will distribute via truss action of the rafters and ceiling joist.
A ridge beam is a structural member that is supports loads, can span, and thus different from the ridge board. The fact that a ridge beam is made from the same materials and looks just like a ridge board confuses most people (this entire thread being a remakable perfect example).
To illustrate my point below, I'll also define a truss - a set of strutural members arragned such that you get primilary axial load (compression and tension) in the members.
When you construct a system that has multiple potential load paths, the forces are distriburted based on stiffness of the load path. In the OP's roof, there is a ridge member (note I didn't say board or beam) runnning front to back. In the perpendicular direction, there are triangles (note I didn't say trusses) formed between the rafters and ceiling/floor members. So gravity loading in the OP's roof has a choice - if the ridge member is stiff enough (ie doesn't deflect to activate a different load path) and its ends supported to carry the load, the ridge member can be a beam, supporting the rafter ends at the peak. In this force distribution, there is only bending in the ridge member, bending in the rafters, and any loads applied to the storage area would induce bending in the floor joists. Alternatively, if the ridge member is not as stiff as the trianglar geometery of the rafters and ceiling joists forming a simple truss, the ridge member won't span at all and load will distribute to the rafters/ceiling joists with truss behavior. In this force distribution, you get all of the bending in the rafters and ceiling joists that you had with a ridge beam but you also get axial forces - compression in the rafters and tension in the ceiling joists.
The vast majority of gable shaped roofs are oriented such that their ridge is the long dimension making it nearly impossible to construct a deep enough/stiff enough ridge member to be stiffer and span relative to the rafter/ceiling joist truss system. The OP's building is square but ultimately has the same issue. He has a ridge board that can't span and all load will be transfered via the trusses created by the rafters/ceiling joists interconnections. Rafters will therefore have combined axial compression (from the truss action) and bending (because it spans between the ridge and ceiling joist connections). The ceiling joists will have combined axial tension (from the truss action) and bending (because it spans between the rafter ends, center vertical hangar, and other rafter end). The rafters will probably be fine but the builder put a splice in the ceiling joist right at maximum bending moment (for a two span member) that also has tension. Wood has like 1/2 to 1/4 capacity in tension compared with compression. I'm not about to go and do a structural anslsyis of all the members, but it is clear that the weak spot in that roof is the center splice in the ceiling joist.