With the info posted here, I did an experiment.
The large bar is what locks the door in place into the strike plate.
The smaller bar should not fall inside the strike plate, but rather be
depressed about half-way. When it is, it prohibits the large bar from
being depressed into the door. So to answer my original question, I
think the door should fit snug just as the large bar extends into the
stike plate, and at that time, the small bar should be just behind the
large bar, but resting on the strike plate and not inside the opening
of the strike plate.
So back to my real issue. My door is not fitting properly. I think
it's hung slightly out of square as the lock side bottom is lower than
the hinge side bottom, and the crack along the lock-side is not
equally spaced. When closed, I can see daylight towards the bottom of
the lock-side edge, and when it rains I get some water coming inside.
The sweep at the bottom also rubs too much on the lock-side (as
compared to the hinge-side), and is worn down and cracked and split.
So what I really need is an education on how a properly hung door
should fit, and then how to hang the door in that way.