About transformers: very little primary current if the secondary is unloaded
Yes, you are correct. A transformer always has a "magnetizing current" (reactive) that is determined by its own inductance, whether or not it is loaded.

Loading the secondary imposes a second "resistive" (in-phase) current that varies with the load.

It sounds as if the book you quoted is referring to this in-phase component only, and ignoring the magnetizing current.