I'd hazard to guess most people treat these things as just regular English words or phrases which have diverged in meaning, not even realizing it is/was Latin with a specific meaning. Particularly e.g./i.e., I remember them by the English translations and couldn't even tell you what the Latin they stand for is without looking it up.

I think the author used the word correctly but explained the circumstances somewhat confusingly.

IIUC an entity engaged a service to register a domain on its behalf. So the entity did indeed register via the service, but it would also be correct to state that registration was carried out by the service. That service also happens to provide a link shortener. So instead of referring to the service as "company X" the author went with "link shortener".