In the sense that people with rhinovirus symptoms usually say "I have a cold", yes.
It seems it's best to think of "cold" and "flu" as totally informal terms with a wide range of possible meanings. I've had Actual Real Influenza twice and both times it knocked me on my ass for a week, lost weight, etc. yet I hear people saying "I have the flu" when they have mild symptoms and only miss a single day of work. Hard to take such a wide range medically seriously.
It seems it's best to think of "cold" and "flu" as totally informal terms with a wide range of possible meanings. I've had Actual Real Influenza twice and both times it knocked me on my ass for a week, lost weight, etc. yet I hear people saying "I have the flu" when they have mild symptoms and only miss a single day of work. Hard to take such a wide range medically seriously.