In my audio learning, I hear the beginning consonant sound of the same word spoken by two different speakers to be as a "w", as in "wari", or to be an "m", as in "mari". While it seems clear that the Japanese have a different sense of this sound, it proves a little difficult to distinguish which pronunciation one is supposed to be hearing in the first place. What simple rule can I apply to sort out this sound when I hear it?
While it seems clear that the Japanese have a different sense of this sound
What you're actually referring too is called the phoneme; a discrete contrastive unit of sound of a language. In IPA transcription, "W"=[ɰ], and "M"=[m], where the bracket notation indicates phonetic transcription. These two are indeed part of the phonemic inventory of Japanese, meaning that if a Japanese speaker were to use one where the other was called for, the result would be either a mispronunciation or an entirely different unintended word. It is most likely that you are misperceiving one of them, or it is the fault of the recording or machine. Look up the two permutations in a dictionary, probably only one will be listed, unless the word is in fact a suffixed word.
If you had provided the words to which those supposed sounds belong, someone could have disambiguated the intended sound. Given the intended sound and word, one could obviously write down the so-called rules that were involved. There are of course rules, linguistic rules at any rate, such as, for example, that one phoneme cannot neighbour another phoneme in the speech stream.
What simple rule can I apply to sort out this sound when I hear it?
What is the sound being sorted against? I'm not sure what you mean. Perception strategies for another persons speech? Acoustic cues to the articulated phoneme?
I guess to answer your actual question "Is this consonant...inflection?". There's different uses of the word inflection. Morphological inflection in grammar is one, and the other commonplace usage, I think, is a misnomer meaning "change in intonation" or something to that affect. I hear this word all the time from the common folk, and I even used to use this word myself, but I've never seen the the word "inflection" in the context of phonetics/phonology. Looking at the wiki for Infelction it is entirely the morphological sense. I think it's just a misused word in the daily vernacular. For some reason people think it means change in pitch (as opposed to the well-known fixed tones of certain Chinese dialects) or something. I have no idea. You meant inflection in this sense of changing pitch, right?
The answer then I think is no; intonation (the pitch profile) acts independently of the phonemic form of a utterance.