What is the correct term for converting data between client side and server side format?

There is no term for converting data to and from a client-side format to a server-side format, mainly because there doesn't exist such thing as server-side format and client-side format. Depending on the languages and code conventions used on both sides, data can live in very different formats and there is no server-side format that encompasses all server-side data formats, likewise for client-side data formats.

Case conversion is just a matter of code conventions in the server and client side. I've never encountered a term that encompasses the concept of case conversion specifically in the case of server-client communication.

So you'll have to invent your own term here. I'd suggest client-server data mapping.

I would call what you're doing mapping, since you're rearranging your data and renaming variables.

Serialization is the process of converting an object to a string or a "serial" format like binary, but you're doing more than that.

I'm not sure there's a single term for this. Snake case versus camel case usually involves programming constructs, not data manipulation, and is rarely a server versus client issue.

The only term that comes to mind is "canonicalization", which means converting data into a standard format. This is often done when allowing flexible formats of data coming from the client, but allowing all the modules that process it to be written to expect a more restricted format.

For instance, the back-end code might only understand camel case, but the UI or API allows both snake case and camel case. The main entrypoint would canonicalize everything to camel case.