Yes, some alcohol burners have the primer pan attached, and on others it is separate. In some cases it makes a broader platform for the burner.
I've watched my own burners, and when the fuel is cold, it goes something like this: You ignite the main load of fuel, but it barely burns for a short time. The vapor is lit, but there isn't much happening. Then after 30 seconds or so, it starts to pick up a bit. Then a little more. It takes the darned thing maybe a minute or two just to get the burner going to what I call a normal flame. Then it does the normal boiling job, and then it runs out of fuel.
Alternatively, if I prime it with a very small dose of alcohol, only that primer burns first, and it seems to warm up the main load. Then I ignite the main load, and it seems to get to normal boiling very fast. I don't see that there is much difference in total fuel used, but the time is better.