If a fuel doesn't provide more energy than it takes to get it - then you are using it (hydrogen) as a (poor) energy storage system. Otherwise, what would be the point?
Whether or not some material has an EROI that is positive or negative when used as an "energy storage system", is an economic question, not a question for chemistry and physics.
When we have to resort to citing dictionary definitions, this is generally a strong indication that a discussion/debate has passed the point at which it's useful, but perhaps this is an exception to that rule.
fu·el
noun
material such as coal, gas, or oil that is burned to produce heat or power.
* * * * * *
There's nothing there at all about whether or not more energy has to be added during the generating process than is present in the fuel.
Would you bother heating your house with wood, if it took more energy to get the wood, than you got out of it?
It depends on just how useful the energy is when it's released. If it's highly useful, then it might well be worth it.
For example, the first economically workable steam engine, the "Newcomen Atmospheric Engine", was only 0.5% efficient (one-half of one percent efficient!) at converting the energy in coal to useful power. It unquestionably took more energy to mine that coal and move it into the Newcomen Engine than the engine was able to harness when it was pumping water out of the coal mine.
Did that mean the coal burned in it was "not a fuel"? Of course not! It just meant it nearly all the energy was wasted. At least the well-to-wheel process of using hydrogen fuel to power a fool cell car, poor as it is, is more efficient than that!
If aluminum air batteries are not actually batteries, then what are alkaline batteries?
Battery cells.
I think I should step aside and let you argue with the dictionary.
-