Actual dictionary definitions: Doing some quick dictionary searching for "arbitrary" gives the definition: "based on random choice or personal whim, rather than any reason or system." The definition given for "random" is "made, done, happening, or chosen without method or conscious decision." In fact, "random" is listed as a synonym for "arbitrary" on an online dictionary. Therefore the interchanging of the two terms is completely understandable.
Technical vs natural language: While it is true that "arbitrary" vs "random" have very different technical meanings in mathematics, they are nearly interchangeable in natural language. It is important to distinguish between natural and technical language usage/meaning.
I would explain this distinction between natural and technical language to my students. That is something that is students of any discipline should be aware of. There may be a risk of muddying the waters though since mastering the actual mathematics at hand may or may not be helped by this discussion of language.
Are you doing the "right" thing in correcting them? Offering a student relevant correct information is always the "right" thing to do. However, it may not always be the right thing to do if there is sufficient risk of it causing more confusion.
Language is a huge problem in mathematics. It's not something that is taught well in my opinion, in terms of how to actually speak mathematics. At least, if the way my students talk is any indication, there is generally a huge gap in being able to do mathematics and being able to explain it verbally in a coherent fashion using technical terminology correctly.
The physical process of choosing a number: Now let's consider the actual physical process of a human choosing an arbitrary number (in the technical sense here). It might be the case that such a physical process could be modeled using a random variable.
So, inasmuch as it is a real process of coming up with an actual example of an arbitrary number, it might actually be a type of random number in the probabilistic modeling sense. Of course, in the actual mathematical context where the number is to be used, it is just an arbitrary number, e.g. to be plugged into an equation.