“Self-esteem is reliance on one’s power to think.” It's not clear whether this is meant as a definition or as a statement of fact. Objectivists often confuse the one with the other. A definition merely defines how a term is used. One may define one's terms as one pleases, but once a definition is granted, one needs to stay consistent to the usuage. If Rand's statement was meant as a definition, she is guilty of equivocation; for she does not always stick to that particular usage.
Self-esteem, in Rand and other writers, generally means esteeming one's self. Whether such esteem is based on a reliance of one's power to think is an assertion about matters of fact that requires evidence. Such evidence that is commonly available suggests that there is little relation between personal achievement (whether in thinking or in other areas) and self-esteem. American school children test highest for self-esteem and yet rank among the bottom in academic achievement. Criminals also test remarkably high for self-esteem, although their powers of thought are often sub-mediocre. Sometimes it is precisely who think well of themselves who lack the motivation necessary to become powerful thinkers. Why should they? They think themselves great as it is. Why bother with self-improvement if you already think you're great? On the other hand, there are those whose very lack of self-esteem serves as a motivator for self-improvement.
“Only a rationally selfish man, a man of self-esteem, is capable of love.” This statement packs three assertions, any one of which could easily be dismissed on empirical grounds. It assumes that, in order to be capable of love, one must be (1) rational, (2) selfish and (3) a man of self-esteem. Does Rand provide any evidence of these assertions? No. Indeed, they are hardly plausible. If Rand's view was true, we would have to conclude that most people are incapable of love. Would any sane person actually believe such a thing?
“Humility is not a recognition of one’s failings, but a rejection of morality.” So are humble people immoral (or amoral)? Again, we are confronted by a grossly implausible statement asserted without a jot of evidence to support it. Worse, Rand seems to be trying to redefine humility without making it entirely clear that she is doing so. Rand's tendency to redefine terms, not merely for herself, but for others, constitutes an egregious intellectual vice. She is, in effect, putting words in other people mouths and then condemning them on that basis. If she wishes to redefine humility, then she should do so in forthright terms, with a complete understanding that her usuage of the word has nothing to do with how the word is used in common discourse. And when chooses to use her redefined term in some controversial statement about matters of fact, she needs to back up her statement with factual evidence. Redefinition does not constitute proof.