I'm only aware of F1 , F2 ect as used in general genetics to keep track of subsequent generations from the parental generation ... as you already mentioned. An "X" simply means a cross between two plants ... I'm not a plant breeder .. I'm sure there are some standard conventions of symbols / nomenclature between breeders but F1 , F2 means simply what you said it does... a method to number and keep track of generations. Who came from who ? .. or who was mixed with who ?

So ... if you had a wild onion "you" may decide to call it Wo and if you crossed it with an Onion stock resulting from a cross between two initial onion stocks it could be denoted for your purposes as Ow X F1 ... meaning a wild type onion crossed with the first generation of an intitial cross between some starting parental stock.

Now if you end up with something new and different you can retrace and make some sense of your breeding experiments not only to duplicate your results but to understand what genes are responsible for those traits.

Again I don't breed plants and was never an ace in genetics ... Eeek.

Good Day ...