In order to avoid sounding repetitious by too frequently using the word "word," I would like to be able use the word "expression," writing, e.g., "Applying the function $f$ to the lang$_{1}$ word $ab$ yields the expression $abba$, which is a word in the lang$_{2}$ language."
Is that an acceptable usage of the term expression?
Also, if I were to instead write "Applying the function $f$ to the lang$_{1}$ word $ab$ yields the word $abba$, which is a member of the lang$_2$ language," could I be criticized for first identifying $abba$ as a word without specifying the set in which it is considered such until I reach my sentence's dependent clause to reveal that crucial piece of information? Or is it okay to call something a "word" in the general sense that there exists some language in which it is a word? (I know I could rephrase using "...the lang$_{2}$ word $abba$...", but such rephrasing would prove extremely awkward, given the context of my discussion.)
Expression isn’t actually wrong, but it’s awfully general; when you’re talking about a word, it’s generally better to use the more precise term word (or sometimes string). For the example in your first paragraph I would write simply:
Your version is unnecessarily wordy. There’s no real need here for function, word, or expression: all are perfectly clear from the context.