Teg starter problem

The starter in our 98 integra is intermitant, it will work when the battery is up, but will be totally dead if the battery has set a long time or during pit stops in an endurance race. Then it is suspect; Either pulls in with delay or not at all.

This is a generic rebuild starter about 2 years old, I replaced the battery this fall and no improvement. I did notice both before and after replacing the battery that it won't pull in if the headights and heater fan is on. Turn them off and the starter works. I hate to just guess and throw a new starter at it . Has anybody seen similar starter symptoms to what I am seeing and identified a cause?

tks
bob

Update - It appears that the terminal plug connection had gotten loose. We tightened the terminals and it seems good now. Thanks