Thanks for notifying me about your posts. I’ve added my email address to the relevant post.
You ask if ‘Natural Argumentation’ is the same as ‘Informal Logic”.
There’s no consensus on this matter but, as I use the terms, natural argumentation refers to all the modes of reasoning we employ before undergoing logical training. Informal Logic, on the other hand, is a term I use to signify that part of our natural reasoning which is not amenable to formal analysis.
Re your second question, whether the notion of inference applies only to formal logic and not informal logic:
The term ‘inference’ comes from the Latin very meaning to carry or bear, and the basic sense of the term is that if one starts with certain premises which are true, the truth is ‘carried over’ to the conclusion. Inference, properly speaking, pertains to formal logic but it can be used in a weighted sense (i.e. more or less, to a greater or lesser degree, etc) in informal logic.