I think this misunderstands what a "cold" is. Merely being a respiratory infection doesn't make something a cold. We call something a cold when it's 1) a mild upper respiratory infection that 2) is viral, but 3) is not specifically attributable.
There are colds caused by influenza viruses, but because "the common cold is defined on the basis of its clinical presentation, a mild influenza infection can accurately be diagnosed as a cold, meaning that the two infections are not completely distinct disease entities."[0] Once someone with a cold tests positive for the flu, though, we would generally correct the description to a mild case of the flu and no longer say they have a cold.
There are colds caused by influenza viruses, but because "the common cold is defined on the basis of its clinical presentation, a mild influenza infection can accurately be diagnosed as a cold, meaning that the two infections are not completely distinct disease entities."[0] Once someone with a cold tests positive for the flu, though, we would generally correct the description to a mild case of the flu and no longer say they have a cold.
[0] https://www.hmsreview.org/issue/2015/1/the-most-common-illne...