Wrap the_title_attribute in a H2 within a Conditional Tag?

The the_title_attribute() template tag is intended specifically for outputting the Title formatted for use in an anchor tag title attribute. The the_title() and get_the_title() template tags are intended for general print/display of the Title.

I would recommend using get_the_title(), rather than the_title_attribute(). To wit:

elseif( get_post_type($post->ID) == 'press' ){echo'<h2>'. get_the_title() .'</h2>';}

I believe this would be more semantically correct.