The flu is the influenza virus, period. The common cold is typically caused by a rhinovirus, but coronaviruses, adenoviruses, and influenza viruses can cause colds too.

But note that “the influenza virus” is a term that includes many genetically distinct mutations.

Sure, and there are around 160 known types of rhinovirus, 50 adenovirus types, and now 7 known forms of human corona viruses. Flu refers to a specific family of RNA viruses, common cold is a much more broad term.