Here is an article about a guy who ran a test using scent to see if his dog was self aware. It makes for interesting reading in relation to what I was saying about bias, and also at least partially, answers your question about one way we could better test animals for self awareness.
https://animalwise.org/2011/08/16/the-yellow-snow-test-for-self-recognition/
This seems to me a much fairer way to test various animals. Test to their strengths, rather than testing them to our strengths.
It is a topic of great interest to me. If we start from a position in a religion that mankind is special and God places us above all other life, we will automatically draw one kind of conclusions.
But what if we were to come from a point of view that man is just another animal. Not saying that a God could not have been involved; rather that said God didn't mean for us to be any better than any other life form on the planet.
For me, now this gets very interesting! We humans are definitely a breed apart from any other species on the planet. If only for our ability to alter it specific to our own needs and desires. But if it is a random mutation, does it mean our species is a good one for the planet? It is hard to look at what we have done to our only home and suggest that we have been good for the biosphere; good for our fellow creatures. It could turn out that the mutation that made us humans might be the worst thing to happen to this planet. We have the capability to destroy it for almost all life for a very long period of time.
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The other part of this equation for me is the randomness of evolution. For a long time it was thought that life progressed forward towards an ever, more capable set of species. With humans being at the apex. But we now know that evolution is a truly random role of the biological dice. It has brought along better species that have failed, and lesser suited species that have succeeded. The mutation for humanity could have happened much earlier in the life of the planet.
The 'common knowledge' that the rise of the mammals was due to the extinction of the dinosaurs has been proven to be false. In fact mammals were around for as long as there were dinosaurs. A version of early mammals were around even before there were dinosaurs!
The Mammal-like Reptiles, or Therapsids first appeared about 285 million years ago near the begiining of the Permian which is well before the dinosaurs. They evolved quickly and many different groups arose. They were very successful until about the end of the Permian, about 245 million years ago, when something catastrophic affected the earth and nearly all of the species then living died out. New species evolved rapidly to fill this empty habitat, among them the first dinosaurs and a few million years later the first mammals.
http://www.earthlife.net/mammals/evolution.html