Standard relays do not normally have internal diodes, except for some with a diode across the coil. The ones you have illustrated would be special purpose relays. Relay A would switch current (in either direction) when the contact (shown as open) is closed by the relay, and the diode is not in the switching logic, just present to use as needed. Relay B is similar to A in switching, but the diode is directly connected to one switch contact; if, for example, source power (+) is on the left side of the switch, current would flow to the other contact and also through the diode. Because of the special nature of these relays, they are not interchangeable.
Cost of a common relay depends greatly on its features; things like, is it weather-sealed, how much voltage and current can it switch, reliability figure of merit, etc. Assuming it is weather-sealed, and switches a nominal automotive-rated voltage at 10A or less, then I'd guess less then about $20 each. It's been a while since I ordered sealed relays, and they were mil-spec 28V types, which cost lots more.
Customized relays are not uncommon; if you need a diode in the circuit for control, where do you put it, otherwise? It does create problems, years later, when the custom types become hard to get, and we have to improvise. The same would be true for a diode inserted into the harness somewhere.
so I always thought of the electrical system on a motorcycle as a specialized part of it that is not interchangable with other machines and usually has been customized in some way that makes it impossible to alter without destroying... but....
I will know alot more after I get my outdated, but supposedly correct shop manual... yes I forgot how to spell payshunts.
Probably, unless the Yamaha bolts are an unusual size. Sometimes, a bolt will be longer than normally comes in a given diameter and thread. Assuming that is not the case, I usually go for a higher grade in the same size.