I've also been noticing that there are a bunch of notions of "rationality" being thrown around. This is because "rationality" is used to describe different domains of thought, and in each of these domains, the term has fundamentally different meanings (some of which you were getting at). Here are at least some of them:

Deductive rationality: This one is relatively easy. When talking about the rationality of a deductive argument, we mean that the argument is valid (the conclusions follow from the premises), and we often also mean that the premises are true.

Inductive rationality: This type of rationality governs empirical claims. There a various controversies about when inductive arguments are rational or not, which I won't get into but are well-explored in the philosophy of science.

Moral rationality: What does it mean to say that a moral argument is rational? Having the argument be deductively valid would be a necessary condition, but might not be sufficient. Whether you are a moral skeptic or not, the truth conditions for moral prescriptions are different from the truth conditions of descriptive claims about the world.

Pragmatic rationality: This is the rationality of an agent's actions relative to that agent's goals.

Lack of emotion: If someone is being emotional, it doesn't necessarily mean their arguments lack the forms of rationality above. This is the one usage of the term "rationality" that is arguably bogus, since rationality is at best moderately correlated with lack of emotionality.

In short, when we say: - "Your mathematical proof is rational" - "Your scientific theory is rational" - "Human rights are rational" - "Your decisions are rational" - "Calm down and be rational!"

the notion of rationality invoked in each of these statements is different.

But how different? Is there a generalized mode of thinking that leads to rationality in all these domains (except, perhaps, lack of emotion, since I think that a bogus notion of rationality)?

Deductive and inductive reasoning are indeed distinct, yet with moral and pragmatic reasoning, once you accept certain premises about the rightness or effectiveness of certain actions, then those problems break down into deduction and induction.

Deduction and induction themselves have similarities, in that both involve the consistent application of rules and principles, though in the case of induction, those rules are more complex and debatable. So your definition of rationality, "having explicit and conscious standards of validity, and applying them in a systematic way," is at least necessary, but not quite sufficient. As others have observed, more elaboration on the actual nature of those standards of validity is necessary. For example, a religion might involve explicit/conscious/systematic standards of validity, some of which come from divine revelation. Yet we should hesitate in labeling it rational, since those standards are suspect, because the divine revelation they rest on fails to pass explicit/conscious/systematic standards of validity.

A good start would be to make your requirement for application of explicit/conscious/systematic recursive, though then we would need a way to avoid infinite regress.