Example 1
As they walked through the abandoned hospital, Craig and Meryl could feel a ghostly presence following them.
Worst. First date. Ever. Since it sure sounds like Craig and Meryl aren't alone, and someone, or something, else is present in the run-down hospital, presence is the right word for this spooky sentence.
Example 2
On the first day of class, Ms. Larue presents each of her new students with a monogrammed fountain pen.
It's an elegant gift, sure, but not entirely practical given that Ms. Larue teaches gym. In this example, presents is a verb that means to give.