A few days ago I read this question on the Biology Stack Exchange
Does cheese contain cells of animal stomachs?
It reminds us how rennet is made:
Dried and cleaned stomachs of young calves are sliced into small pieces and then put into salt water or whey, together with some vinegar or wine to lower the pH of the solution. After some time (overnight or several days), the solution is filtered. The crude rennet that remains in the filtered solution can then be used to coagulate milk.
I think to key to the definition of vegetarianism that excludes rennet, and is, I think, the definition most widely used among vegetarians in my country (the UK), is in the first sentence.
I think that definition of a vegetarian is does not eat the substance of the body of a dead animal, or perhaps does not eat substances obtained as a direct result of the death of an animal.
Your question draws attention to the fact that other substances obtained from animals cause death:
I've seen in some places people alternatively describe vegetarianism as "not using/eating things that entailed an animal's death" (although that would seem to preclude milk from being vegetarian)
I assume that you are referring to the male calves of lactating mother cows who are killed because the dairy industry has no use for them. Male chicks are also killed at birth because there is no use for them in the egg industry.
We can see the difference between the two definitions. In the operative one that excludes rennet but not milk, we won't eat the dead body or its substances. We will not eat the dead themselves. This is true even if the substances themselves have been removed leaving only traces (thinking of non-vegetarian wine). In the one you proposed that [ought to] exclude[s] milk, we won't eat the substances whose production entails other deaths. We will not eat the substances produced only at the cost of death.
The smallness of this difference, the realization that eating the dead one barely differs from eating that which causes death, is one of the paths towards the practice of veganism, but to exclude rennet we don't need to travel that far. In the UK, cheese not made from animal rennet is labelled vegetarian, although some vegetarians here are not strict and eat any cheese and products containing cheese like basil pesto.
Other posts that discuss trace ingredients and challenge definitions and their boundaries: