No, it's just that you live your life with an understanding that carrots are orange not purple. Not based upon emotion, but internalized completely.
It is not a "heartfelt belief" that makes the carrot orange. It is truth; it is fact. The carrot, regardless of what anyone says, is orange. Satan may convince someone it is purple, but the truth is, is that it is orange. "Feelings" cannot change that fact.
You have illustrated my point...."feelings" don't make things right. "Truth" makes things right. Truth is truth, forever. I'm assuming here that you and I are both Christians. If so, you and I would agree that killing a Jew because he/she is a Jew is wrong. That is a fact based on our belief, morals, and Biblical teaching.
When a Hitler comes along and says Jews should be killed because they are an abomination to the human race, and we've been screwed by a Jewish person, we may "feel" like the Jews should be killed. We may even be "heart-felt" in our belief because of growing up in an anti-Semitic house, with a father who may have been laid off by a Jewish owner in the depression, listening to anti-Semitic radio programs. However, the truth is that God did't say any such thing. "Truth" may coincide with "feelings", or it may not. That is why it is so important to us Humans to be able to discern "Truth", with the help of Scripture AND the Church. We should not trust "feelings".
When you say that carrots are orange due to the fact they are orange, what you are stating is a belief. Yes, we can judge your belief for ourselves with observation, but there is no way for us to judge how strongly you hold that belief internally without devising a test to determine your belief level.
The fact that you strongly hold the belief has nothing to do with your emotional attachment to the facts, nor with any emotionally feeling you might have for carrots in general. But we could probably determine that you in fact do "believe" this belief and live your life with no question in your mind that you are right. This does not make your belief an emotional one, but many might refer to it as "heartfelt".