What you describe (act, schedule, delegate, delete) looks like 'ranking' to me. Under 'classification' I'd understand a seperation without ranking entities. And that at least falls under the request of OP (whether it satisfies it or not).
What I don't understand about your comment is the similarity to your scheme that just has the 'cost' dimention addionally. The 'cost' doesn't make the entities any more 'rankable' though. So why wouldn't I apply your critic to your own proposed system?
This comment is not in bad faith. I'm just a little confused about your point. If you have some sort of experience or qualification in this field, I'd be happy to be corrected.
With that process (the "Eisenhower") you distribute items into four classes; the classes are in a way ranked (between each other), but the items within each class are not sortable with that method. You obtain buckets full of "equal" things.
(And the problem of the OP was having all items under the same scalar of importance. The decisor using the Eisenhower method will obtain a distribution in two sets - "act" and "schedule" - but will still be at a loss about what to tackle first.)
> The 'cost' does[...] make ... rankable
It does - it makes the "ToDo" set of collection of fully ranked items (more precisely, it contributes to it with the other two parameters - it adds insight and complexity with a new dimension).
Take a set of items, build a function of (urgency, importance, lightness), attribute per each item a scalar to each of (u,i,l) and a scalar ("priority") is returned which allows for fully sorting the set of items.
> If you have some sort of experience or qualification in this field
Eisenhower did not say it was his idea. See https://quoteinvestigator.com/2014/05/09/urgent/
> It is a great procedure
That is a classification procedure¹, not a ranking procedure. It does not satisfy the original poster's request.
¹(Acted upon immediately; scheduled for action; delegated; dropped.)