Well, to actually call yourself a bayesian, some may say that you have to explicitly use Bayes theorem to do the updates. To avoid confusion, you may wish to use a more accurate term. Around here we use them term "rationalist" in the sense you were using "bayesian", and more people will understand you if that is the word you use. Ultimately, it's just a question of words, but you do want to avoid confusion and have people understand you, so it is a good idea to use words in the style others use them.