In the context of a modern PC :
Is it correct to say that these programs are all examples of Application/User programs that need system calls to request OS services for performing privileged operations like writing a file?
Has the term Application the same meaning as User program has ?
A lot of these terms are only defined through common usage, there's no official definition.
A user program is the most general. A program was the term for a sequence of instructions a primitive computer would carry out, by analogy to program meaning order of a performance or a group activity or conference. It was a single unit of various special purpose activities grouped together in a useful sequence that accomplishes something. Originally there were only users of a computer, but eventually roles were separated into administrators and support, and those who just used the applications for something. Programs written for those were user programs.
Programs have purposes, and some are small and do a specific thing, others are larger, possibly interactive, are more flexible, and may be interactive. The former are generally thought of as utilities (or tools), the latter are applications (applied to non-computer oriented tasks, like sales entry or word processing).
Some applications are complex enough that they come in parts, or have plug-ins, and some can run entire applications within. Web browsers are an example, and the applications they can run were called applets, as in smaller applications, with limitations from the applications they run within.
Smart phones generally have limits on the applications they can run, so they're not full computers. So the applications are closer to applets, with limits imposed by the store or the OS environment. But they're not run within another application, so aren't "mini applications" like applets. I think "app" is just a shortened version of application, like language tends to do (as "telephone" got shortened to "phone").
Your C program, if it's something you run stand alone, might be a utility or an application, but either way it's a user program. You could also write plugins, OS modules, libraries, and other things that are not programs.