It is a critical distinction.
I am very pro-animal-welfare. I am very anti-animal-rights.
Welfare = humane treatment, consideration for the animal's comfort, needs (including mental/psychological), etc.
Rights, just like for people, would be giving animals things like fair trials, religious freedom, and the right to vote. I'm not sure how we do that.
Freedom from human intervention, while it sounds nice, isn't actually possible in a lot of ways.
Domestic animals would die without human intervention in most cases, as they aren't able to live in the wild.
Many wild animals would also die without human intervention, since we've rather screwed things up. Sure, if we evacuated an entire continent, eventually things would balance themselves back out again. As that's unlikely to happen, and we still have global warming, etc., it seems humans will be interacting with wild animals, too.
Certainly, in some instances, the interaction should be about limiting human's impact on wild habitat. However, in other cases, it's about using human influence to maintain the balance as best we can - like controlling non-native species, or providing the effects on the environment that we removed when we removed a native species (such as when we remove a predator, and then a prey species overpopulates, and we risk the prey species eating plants out of existance and out competing others, and having disease outbreaks, starvation...etc. etc.)
A holistic (holistic meaning to treat the whole, please note that holistic as a word has been coopted by the alternative medicine community) approach to ecology is the best approach.