Why we called al2o3 is lewis acid

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Aluminium oxide is amphoteric. It is easy to see that it is a Bronsted-Lowry base through the following reaction:

Al2O3+6HCl →2AlCl3+3H2O
AlX2OX3+6HCl →2AlClX3+3HX2O
The Alumnium oxide splits and the oxygen accepts a proton, forming water.

But what about the reaction with a base? In my textbook, they say:

Al2O3+NaOH →2NaAlO2+H2O
AlX2OX3+NaOH →2NaAlOX2+HX2O
Now, the textbook claims that Aluminium oxide is an acid merely because it reacts with a base to form a salt and water, as is characteristic of a neutralization reaction.

But I'm not satisfied with this definition. I mean, acids aren't defined as 'things that neutralize bases', we have well-established definitions for them.

I tried to figure out for myself how this could be. Clearly, the Bronsted-Lowry theory cannot be applied here since the compound in question has no protons to donate. Therefore, the only alternative is the Lewis concept. I cannot see how that is applicable in this case.

The most basic definition of "acid" is that it is a proton donor (or one which accepts a lone pair)

All of this stuff is done in an aqueous medium, so we can assume that all aqueous ions and molecules are present. With this assumption (in this case, we are assuming that OH−OHX− is available to react), we get the following equation:

Al2O3+OH−⟶2AlO2−+H+
AlX2OX3+OHX−⟶2AlOX2X−+HX+
Similarly, we get:

Al2O3+6H+⟶2Al3++H2O
AlX2OX3+6HX+⟶2AlX3++HX2O
where it is acting like a proton acceptor (base).