What is technically considered an attack?

[Unoriginal]

They are "special attacks", but they don't count as attacks.

Yes they do, as they are special melee attacks . Special melee attacks that don't have to obey the general attack rules because, once again, specific beats general.

They are even called attacks (without mention of "special") in the same sentence as the other, regular kind of attacks.

Find me anywhere in the PHB where it says they count as attacks despite not having an attack roll.

If you're able to make multiple attacks with the Attack action, this attack replaces one of them.

"If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack."

There is NO question about if Grapple or Shove are attacks, they are specifically said to be attacks.