I always thought of a virus as purely a "modifier", not having the characteristics of "life" independently. If this was a game, the virus might be a runestone or skin for your character.
Anything that doesn't need external "life" to come alive, I would consider as "life" in various states. Maybe it's in hibernation, or stasis, or dormant but the life is there. Maybe to keep the silly game analogy, this might be the extra character on your roster.
Doesn't almost any biological entity need external life to come alive via, e.g. reproduction or mitosis or what have you?
This is why I said "to come alive" instead of "to be created". The virus is something that just exists but only becomes alive when mixed with something that's already alive.
You just proposed a definition. Good. It's not complete, but necessary, conditionally.
Basically, we shouldn't use "alive" and "dead" as dualities. There's at least three states: "dead/inert/never gonna get there", "meets a definition of life (see next category) when supplemented by a host cell or something else complex and exterior", "can self-replicate and grow on its own, in a friendly environment with sufficient food/fuel/inputs available = life".
Maybe more. But let's stop pretending biology is dead versus alive, because viruses definitely ruin that.
This does not differ that much from bacterial or fungal spores, or even from plant seeds, which can also be almost "dead", i.e. without detectable metabolism or internal changes, even over many thousands of years, until they reach a favorable environment that triggers their revival.
The difference between a virus and a bacterial spore is that the viral particle contains only a subset of the parts of a living organism, so it could never be brought back to life in an environment where nothing is already alive. However, once the virus takes control over many parts of a cell, which provide the functions that it is missing, like the machinery for protein synthesis, the ensemble formed by the parts brought by the viral particle and the parts formerly belonging to the invaded cell, can be considered as alive and distinct from what the invaded cell was previously.
In any case, the evolution of the viruses and the evolution of the cellular forms of life are entangled, with a lot of genetic material exchanged between them, so considering the viruses as non-living is definitely counter-productive, because neither the viruses nor the cellular forms of life can be understood separately.