That's right, filial, a latin word which I think means offspring or son/daughter.... It was adapted by Life Science to show generation. F1, F2, F3, F4 - F4 is said to produce F0 (the original parents) That's why I would only inbreed until 4 generations.
Genetics in general is confusing - even worse when it comes to bettas. Certain physical appearance is not made up of 1 allele (the parts that forms a gene. eg: from the above - BB..... one "B" is an allele) - but a number of them. To simplify understanding genetics is coded, often with letters (like the above example). When crossed bred, these alleles will mutate or combine with other alleles to form other genes (both common genes and new genes/strain)
Allele is more unstable, in the sense that it's not fixed and may combine in various variations. Single alleles would not show physically unless it's dominant over the other gene/allele. (Eg. a dragon needs to have 2 metallic genes to show dragon scale) Where as genes are more stable and would show physically - like dragon scales, color, fin type, etc. ...... don't ask why some genes/alleles are dominant while others are recessive - because I don't know. LOL
*** it's been decades, so I'm sorry if I got something wrong and hope someone will correct me.
Now to answer your typical question; as far as I know fin type genes hasn't been ... concluded.... identified (???). So I'll MAKE IT UP to simplify explanation. And this is totally theoretical.
Let's say a CT is AA since it's more dominant than HM which will be coded as aa. AA and aa are genetic codes that shows each strain/fin type.
When the alleles are separated, the punnet square becomes something like this: ..... sorry, IDK how to make tables here. Please excuse my method of typing.
F1 AA x aa = . A A
. . . . . . . . a Aa Aa
. . . . . . . . a Aa Aa
Aa is a new genetic code showing half of each type. So fry will be 100% half of both. You will have smaller and uneven web reduction.
F2 Aa x Aa = . A . a
. . . . . . . . A AA. Aa
. . . . . . . . a Aa . aa
Theoretically you should have 25% CT, 25% HM, and 50% mixed genes. But in real life both the CT and HM you get aren't perfect and still unstable.
F3 AA x AA = naturally you should get 100% AA genes or CT. But in real life you will still have odd mutations in some of the fry. This generation should be more stable and should breed true in the following generations.
I agree with MrV, fins strongly depends on external factors like water quality, food, general environment, etc. This is why fins are more difficult to pin point in genetic terms - you will have a lot of fry that doesn't concur to theory. But hopefully this gives you a general idea of how genetic works.