How do I correctly implement hook_bootstrap_colorize_text() to alter button colors in a Bootstrap sub-theme?
Well, I really thought I had tried everything before posting. I didn't think I'd be answering my own question, but I'll put it up here in the case it is helpful to others. Here is what eventually worked for me:

&lt;?php
  //Implements hook_bootstrap_colorize_text_alter() to alter Bootstrap button colors.
  function methodstheme_bootstrap_colorize_text_alter(&amp;$texts) {
    $texts['matches'][t('Request Rotation')] = 'success';
    $texts['contains'][t('Notify')] = 'success';
    $texts['contains'][t('Set')] = 'success';
    $texts['contains'][t('Mark')] = 'success';
    $texts['contains'][t('Assign')] = 'success';
    $texts['contains'][t('Report')] = 'warning';
    $texts['contains'][t('Request')] = 'info';
  }
?&gt;


I had tried this earlier, and had no result after clearing caches. I had tried  placing the template.php file in my sites/all/themes/methodstheme/templates folder as well as the sites/all/themes/methodstheme (not the templates sub-folder) folder. 

Regardless of the folder I placed it in, it was still not working - I had no result after clearing the caches initially.

Placing the file in the sites/all/themes/methodstheme (not the templates sub-folder) folder was part of the solution. I had brushed up on the Bootstrap Starterkit file structure, and I was fairly certain the template.php file should not be in the templates sub-folder, so I started trying other things. Using Drush I cleared the theme-registry cache, the theme list cache, then cleared all caches. I cleared those caches again a second time, and viola! My site is now functioning as expected.

Lesson learned - clearing caches multiple times is sometimes necessary/helpful!