I hate to get all sciencey on you, but bettas and fancy goldfish
are in most cases pure breeds. They are line bred to achieve their looks, not hybridized, just like different breeds of dog look different but are all the same species. Most bettas probably do have some
Betta imbellis in their background, but it's usually so far back in the ancestry that it's not even worth mentioning, especially when we start talking about fancy long-finned varieties. Breeders don't usually cross to other species this far down the line because they're going for fancy fins and colours and the "drab" wild-type fish just aren't part of the program.
Flowerhorns are not a species, and probably never will be. There is not set definition for makes an organism qualify as a species; the ability to produce viable offspring is only part of it. It's really pretty arbitrary. As my evolution professor said, "Fill a room with biologists and tell them they can't come out until they agree on a definition of species and you'll have a bloodbath." While we can't define what makes a species a species, something becomes a species when it is officially scientificially described -- in other words, given its taxonomic classification, as per the definition you gave. As I said, because flowerhorns are hybrids, this is not likely to happen; there are plenty of other
naturally occurring species that haven't even been described yet. Wolf-dogs can usually produce offspring too, but they are not and will never be a species
Edit: This doesn't have anything to do with the topic really, but I feel like ranting since I'm on a roll. Just ignore it unless you care about biology
Hybrid
plants, unlike animals, do become their own species much more frequently. This is because plants can tolerate polyploidy (having a base number of chromosomes greater than "normal") much better. Animals generally only carry two sets of chromosomes, with three copies of even one chromosome usually being lethal (except in the case of Down syndrome and sex chromosome abnormalities). However, plants can easy have three or more copies of every chromosome, but with an uneven number they are infertile. When plants successfully hybridize, it is often by getting one set of chromosomes from one parent and by mutation of either the seed or pollen, getting
two complete sets from the other parent. These plants can live just fine, but are infertile. However, they can reproduce by pieces breaking off and rooting and such, and eventually a plant can emerge where the chromosomes are doubled so that it has 6 complete sets.... and voila, it is fertile! This is now a plant that can reproduce with others of its kind, but usually not with either of the parental species, which is one reason it is more likely to qualify as a species of its own. So in the plant kingdom new species can occur literally overnight!