There are no consistent definitions of the terms "option", "argument", and "flag", and there is no central authority within the software development world that could enforce their usage. This happens with much terminology: after 30+ years of using the word "directory", I now have to deal with people using the word "folder" who have been confused by Microsoft's new-speak.
There are different ways that consensus definitions for terms can come about in programming. In the case of "argument"/"option"/"flag", canonical manuals and tutorials for programming languages have helped to enforce usage, as has the terms used in common libraries.
For instance, the things you put on the command line after a command are often called "arguments" to the command, by analogy with arguments to a function call, and this is probably partly because they are called "arguments" in the C manual (hence
argparse Python library also helps to enforce the term "argument". However, I have also seen them being called "parameters".
The term "option" is derived from "optional", which implies they can be left out. The
getopt C library is one use of this term. But there is precedent for "options" that are not actually optional: for example, the
argparse manual states that one can create a "required option" (although it also says that this is "generally considered bad form"). Options are often preceded by a single (
-) or double (
--, long-option) dash, but there are well-known commands that do not require or enforce dash usage for options (e.g.,
dd). An option can itself take an argument (e.g.,
--color=always), or occasionally multiple arguments.
"Flags" are, in my experience, the same as options, but usually do not take arguments themselves and essentially represent boolean on-off switches.
On a broader note, since every programmer has the option to try and look up some standard way of doing things and naming things, but can also reinvent the wheel without much extra cost, naming is never going to be consistent. And once you have documented your code, and it is clear what new meaning you have given to these words by giving examples, those names and meanings might just stick if there are enough people who pick them up from your code.