What is the class of a spell cast from a staff?

In D&D 5e when a character casts a spell from a staff (e.g. staff of a magi), what is the class of that spell? It is important for items like "+1 to DC for this class spells that you cast" from Tasha’s.

Specific example:

  • I am a lvl 20 Artificer.
  • I am attuned to and holding a Staff of Power, and casting Hold Monster from it.
  • I am also attuned and holding in my other hand (let’s say DM would allow my hand to hold all of them, this is just for illustration) all of the following:
    • +3 All purpose tool (bonus to artificer spells I cast).
    • +2 Arcane Grimoire (bonus to wizard spells I cast).
    • +1 Bloodwell Vial (bonus to sorcerer spells I cast).

What bonus to DC will I get from these items when casting this Hold Monster?