What are the ways to create a new process in C?

As far as I'm aware only fork, vfork and clone are used to create new processes in linux. (posix_spawn is not a syscall, but rather a library function which uses one of the syscalls above to create the process).

exec (and its sibling functions) is used to "mutate" a process, changing its memory layout according to the given ELF file.

(the example below is a very simplified version of the process, that leaves a lot of things out)

For example, when you run a command from your bash terminal, it uses fork (or clone). After the fork syscall, a new process is created and its memory layout is exactly the same as your bash terminal (To understand it better, read about Virtualization & COW, Copy-On-Write). Afterwards, the new process calls execv (or a similar function) to execute the ELF given to it by the command.