We need much better definitions for a few population concepts.
Birth rate as defined by demographers is birth per 1000 women, or something similar. This is not helpful for the general population. Instead I propose we use "Average number of children". This is the number you would get if you asked every adult right before they die, how many children they created. Every child will be counted twice, once by the mother and once by the father. This definition has several benefits that I expect will be illuminated in the subsequent debate.
"Overpopulation" is poorly understood. There are two different concepts that this word has come to mean, but we do not properly differentiate betweeen them. Consider a life boat with 2 people in it. Fish jump in the boat with a frequency that can sustain 1 indefinitely. There are some cans of food that allow the 2nd person to live until the cans run out.
According to the definition of "overpopulation" on Wikipedia, this boat is overpopulated because the only way an organism can have a population larger than what can be sustained by the environment is by consuming resources faster than they renew (the cans of food). Let's use that definition for overpopulation.
The other concept is the situation where the population is at the limit of what can be provided for. But first we need two more definitions.
"Provide" and "provide for": means the number of organisms that can survive in the relative short term. In other words, the boat can provide for 2.
"Sustain": means the number of organisms that can be maintained using only renewable means. The boat can sustain 1.
"suffering the effects of overpopulation" is the best I have come up with to handle the situation where the number of people are being limited by the environment. If we change the boat analogy to be island we can see what this means. On this island are 10 people and the island grows only enough food for 10. If the inhabitants produce a child, so that there are 11 humans, 1 will die, and that is suffering the effects of overpopulation.