The question is about "formal" analysis, proofs, etc... What is a correct definition of a formal proof, or a formal analysis? Is there a good way and a bad way to formally prove something. What is the meaning of a a "semi-formal" proof, "semi-formal" argument.
You may want to check the answers to this question. As the question (and answers) point out, it is not rare to find people use "formal proof" to refer to proofs generated using computer-aided means.
However, as some comments to your question also point out, a "formal" proof of something strictly means a mathematical proof showing that all the claimed (security) properties are met. This can be done using computer-aided tools, but not necessarily. The term provable cryptography refers to a major area in cryptography which deals with the main techniques to create such proofs (not necessarily -and not usually- computer-aided tools). See, for instance, this entry in the Wikipedia for some further insight and references.
About "semi-formal" proofs: I guess that it refers to proofs that do not go down to the full detail, and just set out the overall approach and leave some (or all) parts as sketches. This means that while they may overlook some things, such "semi-formal" proofs point towards the right direction to build a formal proof.