I can inform you that I indeed know nothing of animism. However, a brief lookup in dictionaries indicates definitions such as "the doctrine that all natural objects and the universe itself have souls" and "the belief that inanimate objects and the phenomena of nature are endowed with personal life or a living soul". At this point in my life, I can say I don't believe much in animism.
The problem with dictionary definitions is that they force everything in the world into a manageable, pigeon-holed frame which can be dissected neatly with the intellect.
In my experience, animism is not a kind of faith or belief system, though it is often labeled as such by well-meaning but confused anthropologists.
There is nothing to believe... it is just a form of experience that grows naturally from spending time in and being fully present with the natural world around us. I don't need to believe
in the sun's awareness or that butterflies awareness, I just experience
them as aware. This kind of experience was generally very foreign to white westerners when the pilgrims landed on the East Coast of this country. Such experiences to them were reserved for only their saints. It continues to be generally foreign to Westerners.
The concept of someone experiencing the life around them as aware to the western intellectual mind demands an explanation. The most convenient is to stick it in the category of a "belief system."
The idea of things in the natural world have souls is a Christian interpretation of the animist experience.
I don't think that "non-believers can't possibly have respect for the world around them" is true. For starters, in my mind animists are generally not "believers" of any kind. Most animists I have met became animists because of their experiences. They didn't go down a list of belief systems and say,"Oh, I like this one."
Though, I do believe that when you experience the awareness of the life around you can not easily continue treating life indifferently. I don't think that if people in our culture started "believing" in animism that the world would be a better place. I do think, however, that if lots of people in our culture got to experience the awareness of life they would feel more connected to the natural world and would feel a sense of duty to caretake our planet more, instead of consuming resources without stopping.
I think my true problem is that I haven't been granted the opportunity to truly "be one" with nature. I'm a total nature nut, but my lack of a vehicle means I can't even head out camping in the middle of nowhere on weekends. As a matter of fact, I've never been camping in my entire life (which believe me, is the single fact that crushes me more and more every day). I really need to get away, if only for a week, and enjoy some serious one-on-one time with the elements. At least then I could have a more educated opinion based on first-hand experience.
Yeah. Its that one-on-one time from which the animistic experience grows.