Let's look at the staff again (emphasis mine):
STAFF OF THE MAGI (DMG, pg 203)
... Spell Absorption. While holding the staff, you have advantage on
saving throws against spells. In addition, you can use your reaction
when another creature casts a spell that targets only you.
Looking at Fire Storm (for example, again emphasis mine):
A storm made up of sheets of roaring flame appears in a location you
choose within range. The area of the storm consists of up to ten 10
foot cubes, which you can arrange as you wish. ...
That, to me, reads as though you are not targeting a creature but an area. You can use that spell just to burn down a forest if you wish. No real target.
I think that the staff and the spell work the same in that regard.
If you look at a few other spells:
Fireball: A bright streak flashes from your pointing linger to a
point you choose within range and then blossoms with a low roar into
an explosion of flame.
Prismatic Spray: ... Each creature in a 60-foot cone must make a
Dexterity saving throw. For each target, roll a d8 to determine which
It should not work with those, but these:
Finger of death: You send negative energy coursing through a
creature that you ean see within range, causing it searing pain...
Friends: For the duration, you have advantage on all Charisma checks
directed at one creature of your choice that isn't hostile toward
you. The target must make a Constitution saving throw..
Some tricky ones:
Magic Missile: You create three glowing darts of magical force. Each
dart hits a creature of your choice that you can see within range.
or other single target spells that target others at higher levels.
RAW: I don't really know for those, and I don't think there is a concrete answer. Someone should ask the game developers.
My Thought: With these tricky ones, the part of the spell that targets the holder of said staff could be absorbed. Those do specifically say that they target a creature, so they are eligible for absorption. A DM could possibly say that it absorbs the whole spell (including the parts not targeting the wielder of the weapon). As a DM, I'd allow the wielder the chance to absorb the part of the spell that targeted him. I think these just come down to the DM.