You are right, it was set to a simple search. I also didn't realize that that setting, by default, is allowed to be change by the user. So I set it and did one of the same searches again, using your syntax, and returned equal to or more results than without the OR. The following example contains the syntax that I used with Advanced Search Preference on:
(rehosting OR pdmqa) AND (pdm OR Link)
Thank you for your help Binesh Kumar!
