I understand the blur.
By definition, F3 A x F2 B = F1, the source (A & B) having the priority on the generation. And it's an overall notation, a trait can't be "F4" while the plants aren't etc ...
For a trait you will more qualify it as "stabilized", "unstable" (mix of different degree of this trait in all plants), or eventually "stable recessive" (only a minor portion of the plants carry/express the trait, the others not). The degree of stability is never dictated by the generation, but on the work done on it and on the strategies of selections.
Now if the Breeder A and the Breeder B are developing their stuff from the same original specimens (let's say the same batch of seeds of another breeder), the output is a F4. On qualitative side (genetically), a good one made for the long term. The twos different strategies of selection becoming complementary, if the twos breeders have improved the initial state (goal) each generation of course.
Breeding for red sap can be done also on the PTK, if ever you're able to put the hands on this old EUR amateur ibl. Forming an hybrid from twos distant sources presenting the same occurence of "blood sap specimens" is a good thing in the absolute, but to work each ones separately during enough time as isolated lines permit to offer the guarantee to be able to restrict the genetic code on this goal, more easy.
The genetic shock than is happening in a F1 don't necessary follow the initial expression of both parents, it can be "suppressive", complementary but also it can generate a new genetic paradigm from these similar expression : a new trait than is not expressed in both sources. To give an image, the twos "red sap" parents can generate a "violine sap" than is only possible when these lines are crossed together. A little like the flowers industry if you want.
I will personnally work isolated the twos lines and push them in their genetical limits, until the lines are unable to handle more pressure on this trait (it can take a human life for rich strains). Then eventually i will cross the twos to increase the time i can work on this trait, trough an hybrid of them. But it's not a golden rule at all, it's just because i'm in ease in doing it this way. Nothing say than you can't obtain that with a smart backcrossing program or anything else.
A genetician will just search the right marker, and will do the reverse. Locking the trait, then working the line afterward to make it sexy for the given goal or the given market.
Just use the method than offer the more chances of success to your own sensitivity (which drive the selection), without choosing compromises. A compromise in breeding is the best proof than you have not dedicated enough time to the line. And all lines have their own resistance and "timeline" of work.
Hop it help ;o)