I’ve never liked the word “buddhism.” “Isms” tend to be belief systems; “buddhism” is not concerned with beliefs, even understands them as, ultimately, obstacles. Worse still, the word tends to perpetuate the false idea that buddhist practice has something to do with the worship of a man, the historical figure of Siddhartha Gautama.
The alternative approach, which many buddhists adopt, is simply to speak of “the dharma,” an inherently non-sectarian word. “Dharma” simply means something like “the way it is” or “the nature of reality.” It is a much bigger and deeper term than “buddhism,” which is historically and culturally bound. Perhaps we could say that buddhist teachings expound upon and embody dharma in an exceptionally comprehensive way, but dharma is everywhere and ultimately quite independent of buddhism as a concept. Within buddhism, one isn’t trying to become a buddhist but rather a buddha, a fully realized, fully awake being, having completely perfected both wisdom and compassionate skillful means.
So the benefit of using the word is that we can speak of dharmic qualities in people, practices, art, understanding, institutions, businesses etc. that have nothing to do with the narrower entity of buddhism.
The downside is that if the person reading that word is not aware of what it means or how it is being used, it may come across in a manner diametrically opposed to what is intended. I occasionally have used it and then immediately realized that the person I was speaking to must’ve understood it in the same kind of sense I hear phrases like “the Gospel” or “the Word of the Lord” or “the Truth.” In another words, as a sectarian word bound to a particular tradition.
So it’s a dilemma I haven’t worked my way through yet. At the moment I suppose I feel that, as a term for general use, “dharma” doesn’t tend to work so well, unfortunately. So I use “buddhism” (faithful to its Sanskrit and Pali origin, languages with no upper-case letters), and hope that its lower-casedness conveys some kind of distinction at least. (“The tao” versus “taoism” involves exactly the same issue.)