You were doing okay here, well, at least semi-okay.
The word fish, by itself isn’t actual classification, it is rather nebulous classification, meaning it isn’t that well defined.
However, biologists can have divided the term fish into more well-defined classifications, such as jawless fishes (infraphylum Agnatha) vs jawed fishes (infraphylum Gnathostomata), both are infraphylum of the (subphylum) Vertebrata.
From the Gnathostomata, were several classes of fishes, but the most important ones are:
- the cartilaginous fishes - clade or class Chondrichthyes - fishes that have skeleton made of cartilages, like all the families and species of sharks and of rays;
- the bony fishes - superclass Osteichthyes - fishes with skeleton made of bones.
There are two main classes to the superclass Osteichthyes :
- the ray-finned fishes - class Actinopterygii - their fins are made of bony spines that are webbed together by skin;
- the lobe-finned fishes - class Sacropterygii - their fins have fleshy or muscular limb buds, which are referred to as lobes.
There used to be more species of the sarcopterygian fishes, but most of them became extinct, including the Tetrapodomorpha sacropterygians (eg Tiktaalik, Acanthostega, Ichthyostega, etc); the Tetrapodomorpha are still fishes, but were developing skeletal limb-like fins. The only surviving species of the Sarcopterygii are the coelacanths and lungfishes.
There are lot varieties in the Actinopterygii group.
it is from the Tetrapodomorpha, the clade of Sacropterygii, that eventually developed 4 limbs strong enough their bodies weights to move on land, more freely, the earliest but extinct true tetrapods (the superclass Tetrapoda) - the amphibians.
Amphibians can reside in either aquatic environments and terrestrial environments, but as with all amphibians - extinct or extant - and like their fish ancestors, the amphibians must return to the water, when & where they reproduce, and then they lay their eggs in aquatic environments. Hence fishes and amphibians are classified as “anamniotes“.
Eventually the early amphibians would become so adapted to terrestrial existence, that they abilities to either lay their eggs on dry lands, or retained their embryos in their wombs prior to giving live birth. When they can do one or the other, these are classified as “amniotes”, the Amniota is clade of Tetrapoda.