Immigrate, Emigrate And Migrate

I remember once using the word migrate when referring to someone changing country. The person I was speaking to quickly interjected to say animals migrate. So what is the correct way to describe the movement of people? I did a quick Google search and came up with the following:

Emigrate: Leave one's own country in order to settle permanently in another.

Immigrate: Come to live permanently in a foreign country.

Migrate: The movement by people from one place to another.

Migrate also came up with animals moving as well. So it seems to me that only humans do the first two but either humans or animals do the last one. However, I rarely hear the first two, even when applied to humans. It seems everyone migrates, even when the reference is clearly to people leaving or arriving. So this is my summation:

If people are leaving a country permanently, they are emigrating. If they are a permanently arriving, they are immigrating. If you are talking about the movement of people in a broader sense, they are migrating. To put it another way: My parents emigrated from the UK and immigrated to New Zealand. Therefore they were migrants. I am not the sort to study English but that seems to be the correct understanding.

Immigrants arriving in the US. (but emigrants from Europe).