We tend to think and talk about persuasion in several different ways, what might be called schemes of rational persuasion. They are distinct, although they are often in practice jumbled together. Arguably all forms of persuasion could be talked about directly or indirectly in terms of each scheme, but some things are easier to talk about in one scheme than another, and you do find people panning an entire scheme as defective (although this often doesn't prevent them from implicitly appealing to it elsewhere). I'm not sure how many different schemes there are, or even in every case the best way to individuate them, but here are my suggestions for some obvious candidates.
(1) Evidential Weight
Often we think of rational persuasion in terms of fair assessment of evidence, so that the person who is rationally persuaded has been led to a conclusion by proper weighing of evidence. The implicit assumptions of this scheme seem to be that there are things identifiable as evidence, that these evidences are commensurable, that there is a form of assessment that can handle these evidences in a non-misleading way, and that in the long run evidence converges on truth.
(2) Consequential Tendency
An alternative scheme of rational persuasion takes a person to be rationally persuaded if things they come to accept have certain markers indicating that they will have, or are likely to have, or are of the right sort to have, good consequences. Probably the most generally respectable version of this scheme talks in terms of the promise or fruitfulness of this or that view (does it further inquiry by making other truths easier to find, does it give testable predictions, is it replicable), but there are quite a few different versions. It is important to recognize that this is a completely different scheme from evidential assessment; for all evidential assessment as such implies, the evidence could lead us to completely fruitless dead ends beyond which inquiry cannot go at all -- nothing more to test, nothing more to do -- and for all consequential assessment as such implies, following the most promising consequential lines might take you in a very different direction from the evidence. The implicit assumptions of this scheme seem to be that the relevant possible consequences are identifiable with some reliability, that these possible lines of consequence are commensurable in a way that can be assessed, and that in the long run the most promising positions will be found to be true.
(3) Participation in Rational Community
An alternative way people think about rational persuasion is in terms of what places you in a rational community of inquirers, whether actual or idealized, or allows you to continue participating in such a community. In this way of talking there are things rational people do, and things rational people do not do, and whether you are being reasonable in coming to believe something depends on whether you came to believe it by doing what rational people do and avoiding what rational people don't do. The implicit assumptions of this scheme seem to me to be nicely summarized by C. S. Peirce's famous listing of the "logical sentiments" that are "indispensable requirements of logic": interest in an indefinite community of inquirers, recognition of the possibility of this interest being made supreme, and hope in the unlimited continuation of this community. The idea, in other words, is that rationality of a person is commensurable and assessible (so membership in the community can be determined), that it is possible to give this a sort of primacy over other interests, and that, abstracting from impediments like lack of time, the rational community of inquirers converges on the truth.
(4) Authoritative Source
There's a scheme that's different from any of these that is based on authority; but it is a bit tricky to distinguish from the others, in part, I think, because the vocabulary with which one might express this scheme is less developed than the vocabularies for the others. However, I think it can be distinguished. The basic idea is that coming to believe something in a reasonable or rational way involves depending on the right kind of source of information or guidance. In this it is like evidential assessment, but unlike evidential assessment it emphasizes not the weighing or balancing of evidences but the quality of the source; the sources are assessed, not the evidences. From the perspective of evidential assessment, on its own, there is no particular reason why great evidence can't come from a usually bad source; and from the perspective of authority assessment, considered alone, a good authoritative source may well get you farther than balancing the evidence (e.g., the evidence might be easier for someone else to assess). The emphasis on quality makes it like consequential assessment, but consequential assessment looks at the quality of the position or belief in question, whereas authority assessment looks at the quality of sources or authorities. And while a rational community might have its special authoritative sources of information, community assessment analyzes them in terms of their role in the community, whereas authority assessment analyzes them in terms of features they have in their own right. This scheme implicitly assumes that we can identify authorities and assess them in terms of quality, that following authorities who rate highly in terms of these qualities converges on truth, etc.
(5) Internal Coherence
Very different from all of these is the idea that one comes to believe something rationally or reasonably if it makes your overall view more coherent. We often put this in terms of consistency, but, of course, it admits of degree: it could be that you can get a greater amount of coherence by upgrading to a completely different set of beliefs rather than merely tweaking the positions you already have until they are consistent. Likewise, you can become more rationally coherent by turning merely consistent beliefs into mutually supporting beliefs. The implicit assumptions here are what you would expect: that we can get a good assessment of coherence, that coherence converges on truth, etc.
People appeal to all of these in the process of persuasion, and they also appeal to them in defending the rationality of their own changes in belief. They are all definitely different -- (1) weighs evidence, (2) weighs inquiry-relevant tendencies of positions, (3) weighs whether reaching the belief is consistent with having a role in a community of rational inquiry, (4) weighs the quality of sources, and (5) weighs the structural features of webs of belief and practice. They could all five be integrated, but doing so requires a rather robust and developed account of inquiry, and in practice people tend to use them piecemeal. People also tend to shift around among them depending on what is convenient for them at a given time; I suspect that this is sometimes a serious problem, in fact, since it seems to be common without an integrating account this is not really much different from using completely different standards in ad hoc and arbitrary ways.