Although it's probably fine, it's not the same as non-pyrogenic water for injection.
Distilled water is raised to temperature high enough to kill any living organisms.
Being converted to steam leaves behind solids and high molecular weight contaminants.
However, tiny droplets carried by the steam contain low molecular weight contaminants, mainly the byproducts of living organisms.
Water for injection (aka non-pyrogenic water) goes through another step where the tiny droplets are separated from the steam, resulting in water free from these low molecular weight contaminants which can cause illness.
So there is a difference between distilled water and pharmaceutical grade non-pyrogenic water.
Meanwhile I hope the oil in my gear is at least fast food kitchen grade...
But seriously, the more I learn about this the more I'd bet my ass most no name "BAC water" is BS whipped together without any regard for the actual standards. I doubt most even bother using distilled water.