While we're on the topic of linguistic pedantary, strain isn't exclusive to direct mutations from a parent genome. Strains, like much of biological taxonomy, are a human abstraction to make communication of the idea of -- in this case -- "a virus sharing similar properties to coronaviruses that cause severe acute respiratory syndrome" -- albeit this is a very simplified definition for the sake of brevity.

SARS is caused by SARS-CoV-1 and COVID-19 is caused by SARS-CoV-2.

Rather, if we would like to be absolutely correct about these classifications, we would say SARS-CoV-1 and SARS-CoV-2 are both strains of SARSr-CoV (Severe accute respiratory syndrome related coronavirus), which in itself is a species, an abstract concept used to group related organisms into a convenient umbrella term.

There is no "eukaryote" organism the same way there is no "SARSr-CoV" organism. The added "r" was a recent addition when COVID-19 was discovered.

I will cede that I didn't specify this last point, and you were correct to point it out.