I think the truth lies somewhere in between. AFAIK, yeast needs oxygen to multiply. It can store some of that oxygen in a tricky way so that there will be multiplication after there's already no oxygen left, but on the long run, with no oxygen being present, yeast will stop to multiply. This means that the cell count being present continues to live and to produce alcohol etc. As long as there are the right circumstances (food, pH, alcohol levels ....). But this also means that they won't grow till they "fill the volume", instead they grow as long as they have oxygen and food. If one of these is missing, they will stop the reproduction. So we got an actual interest to keep the starting amount of yeast high enough, because multiplication in the beer just works to a certain degree (ie. Until oxygen is used up). In a starter it's different. It's an open ferment with continuous introduction of oxygen, so there's going to be multiplication until the food is gone.
I have somehow the feeling that some yeasts can multiply without oxygen, I did kveik beers with such a ridiculously low cell count and it finished in no time and the slurry was hundreds of times more then what I introduced at the beginning.