He is teaching classes. That makes him a teacher, regardless of whatever else he does. The LW didn't say that her husband is claiming to be a full or even associate professor, he is claiming to be "teaching at college-X".
He is not a "teaching assistant" he is a teacher.
There is no lie here. There may be a bit of dodging the full truth, but what they don't know won't hurt them.
Making up or requiring the use of some title is pretentious no matter where or what you are (including academics). Correcting someone when they don't reveal the full truth of any matter is a bit anal if you ask me.