'Cognitivism' here relates to knowledge (cf.'cognition'). If moral cognitivism holds, then there are at least some moral truths that can be and are known. Here's how the concept of moral cognitivism can be built up:
Suppose one concedes that moral judgments may be true or false. Does it follow that
he commits himself to moral cognitivism? No, for the obvious, almost supercilious,
reason that propositions and judgments may be true or false though no one knows
them to be so. Suppose, however, one concedes that moral judgments may be true or
false and that we are quite competent to determine that they are. Does it then follow
that he commits himself to moral cognitivism? Still no, because moral cognitivism is a
theory about our competence to know the truth of moral judgments, not merely the
admission that we are cognitively competent in moral matters. (J. Margolis, 'Moral Cognitivism', Ethics, Vol. 85, No. 2 (Jan., 1975), pp. 136-141: 136.)
To point the contrast: moral non-cognitivism is the view that there are no moral truths, hence that there can be no knowledge of them. Expressivism is one form of non-cognitivism. For the expressivist moral judgements are not truth-apt - they are not true or false. They are sentences used not to make true or false assertions but rather to express certain attitudes - pro-attitudes of desires or feelings of approval or con-attitudes of aversion or disapproval. (For the record, expressivism developed from the emotive theory of ethics, the idea that moral judgements are not truth-apt but function merely to express or elicit emotions. Attitudes include but encompass more than emotions.)
I think this explanation answers your question 1) and removes your first definition. Hope this helps.