Your inclinations seem correct to me, but I find if very challenging to be definitive without seeing images.
There are use cases where branding adapts - off the top of my head Apple comes to mind. It's either just the name, the name and the apple, or just the apple. The apple icon can have gradients and a "shine" or be a flat singular color. For video, Apple just used the top of the apple icon, not even the entire icon. All this could also refer to MicroSoft and how they adapt their branding. Or FedEx, with its multiple color variations based upon service. (Granted these are well established brands which offers some freedom.)
Multiple variations are not automatically a bad thing... it's more about how different the variations are and what specific aspects are changing. If the core imagery is conveyed in relatively the same manner, there may be no harm in variations.
A "blur" in itself seems counterintuitive to a "logo". But again, without knowing the company or seeing imagery it's hard to be definitive. A "blur" for branding, to me, conveys a company with a lack of focus which is rarely a favorable impression.
I can't imagine how a "blur" is conveyed for something like a single color product brand placement (like a pen or keychain). And if it's conveyed well at such a small size, my inclination would be to use that variation for all usage. I, personally, am not of a mind to make a brand more complicated for some specific usage merely because it's possible. If a variation works well for silkscreening two colors on a t-shirt, then that variation should be used everywhere, as opposed to making the brand more complex merely for web usage. Just because something can be done, doesn't necessarily mean it must be done.