You and jacobusjay are correct in that creativity is the cornerstone of authorship. It seems like you have done a lot creatively to contribute to the outcome of the photograph, so you're most likely an author of the photograph. The photographer would not be an author if (1) s/he contributed no creativity whatsoever, or (2) the photo was a "work made for hire" with respect to the photographer's contribution. "Work made for hire" has a special meaning in copyright law (not just "I paid him money"); I won't go into a full analysis here but most likely these photographs wouldn't qualify. Also, the amount of creativity needed to constitute copyrightable authorship is very slight. One court has described elements of photography that could constitute copyrightable authorship as including "posing the subjects, lighting, angle, selection of film and camera, evoking the desired expression, and almost any other variant involved." So, when asking yourself if you are the sole author, ask yourself if you controlled every possible variable. Did you select the photographer's camera or lens? To what degree did you control the angle of the shot? If there is any variable that could produce an expressive difference (even a slight variation in brightness, contrast, or the like) that is under the control of the photographer, then that photographer could be deemed an author. Again, I suggest you find an attorney to go over your specific situation. In any event, you can make a strong case that you are also an author, which makes you an owner of the photo.
Even if the photographer is an author, you can obtain his/her ownership stake, however, the transfer must be in writing. That means, even if you pay the photographer handsomely, this doesn't automatically award you his interest in the copyright. There must be some form of written agreement to the effect of "I transfer my copyright to you." Your attorney can evaluate whether such a transfer is necessary.