I've just noticed (by pure coincidence, as I was studying both textbooks) that a paragraph of section 5.5.1 in Rovelli's textbook "Quantum Gravity".
Here's the plagiarized passage
This absence of the familiar spacetime "stage" is called the background independence of the classical theory. Technically, it is realized by the gauge invariance of the action under (active) diffeomorphisms. A diffeomorphism is a transformation that smoothly drags all dynamical fields and particles from one region of the four-dimensional manifold to another (the precise definition of these transformations is given in Chapter 2). In turn, gauge invariance under diffeomorphism (or diffeomorphism invariance) is the consequence of the combination of two properties of the action: its invariance under arbitrary changes of coordinates and the fact that there is no nondynamical "background" field.
And here's the passage that plagiarized
Cette absence d'espace-temps fixe sur lequel se déploieraient les phénomènes se nomme invariance de fond. Techniquement, cela équivaut à l'invariance (de jauge) de l'action sous l'effet des difféomorphismes. Un difféomorphisme n'est rien d'autre qu'une transformation qui fait glisser sans « à-coup » tous les champs et particules d'une région à une autre. L'invariance par difféomorphisme est la conséquence de deux propriétés de l'action : son invariance par changement arbitraire de coordonnées et le fait qu'il n'existe pas de champ de fond non dynamique.
Here's DeepL's translation:
This absence of a fixed space-time on which phenomena unfold is called background invariance. Technically, this is equivalent to the (gauge) invariance of action under the effect of diffeomorphisms. A diffeomorphism is nothing more than a transformation that smoothly slides all fields and particles from one region to another. Invariance by diffeomorphism is the consequence of two properties of the action: its invariance by arbitrary coordinate change and the fact that there is no such thing as a non-dynamic background field.
What should I do? Is it a serious issue?