It's arguable whether or not User Stories is the best name for this concept, but it's a little late to worry about that. My advice is to not worry too much about the precision of what is a user story, but concentrate on the spirit.

A user story makes a difference to a user of the system.

It does something a user wants (or avoids something the user doesn't want).

"Have my data available when I log into the system tomorrow" is a user story. "Write the data to the database" is not; it's a technical task.

"Don't crash when the zip code field is left blank" is a user story. "Validate input" is not.

A user story is functional

It slices through the system, involving all the layers and subsystems as needed.

Splitting stories