I have learned that the word דג means any "animal that lives exclusively in the ocean. As an example, we can see All About Kosher Fish
Any sea creature that does not have fins or scales is not kosher,
regardless of whether it is scientifically classified as a fish or
whether it actually resembles a fish.19 This means that whales,
prawns, shellfish, crabs, octopus, lobster, and shrimp are all not
Aruch HaShulchan, Yoreh De'ah 83:5, but see there 6-11 that according to
Maimonides, any creature that doesn't resemble a fish is not kosher
even if it has fins and scales.
Similarly @DoubleAA has pointed out that Kelim Chapter 17 Mishna 13 also uses this definition for tumah and taharah (ritual purity/impurity).
MISHNAH 13. ALL THAT LIVE IN THE SEA ARE82 CLEAN,83 EXCEPT THE SEA-DOG
BECAUSE IT SEEKS REFUGE84 ON DRY LAND; SO R. AKIBA. IF ONE MADE
VESSELS FROM WHAT GROWS IN THE SEA AND JOINED TO THEM ANYTHING THAT
GROWS ON LAND, EVEN IF ONLY A THREAD OR A CORD, PROVIDED IT IS
SUSCEPTIBLE TO UNCLEANNESS, THEY ARE UNCLEAN.
(82) Unlike animals on land.
(83) Even when dead. Hence vessels made of their skins are insusceptible to uncleanness.
Similarly, an עוף is defined as a flying animal (unlike a "flying squirrel" which is a land animal that glides) since it has wings instead of fore-legs. We see that in the Torah as one of the types of non-kosher עוף is translated as bat.
A chaya or beheima would be defined as any animal that primarily lives on the land and moves using its limbs (legs) and could be examined to determine if it has the signs of kashrus.
Note that reptiles, or snails (as examples) would then be a category of their own that would never be kosher. As a result, the Torah does not refer to them specifically as one of the kinds of being that could have kosher representatives. This would be the class of animals that are referred to as
וְכֹל אֲשֶׁר רֹמֵשׂ עַל הָאֲדָמָה: in Noach 7:8. or in Bereishis 1:25. However, since none of them are kosher, they would not be referred to in the halachos of kashrus.
Alternatively, one can treat all land animals that are not beheimah (domesticated animals with four legs), or עוף, or insects as part of the class of חיה just as one treats all sea creatures as דג (see below).
In that case one could say that a snake would have had limbs except for the curse that caused it to lose its limbs. Thus, it is also a land animal.