Think about it like this: What would have to change to call an conscious thought unconscious or an unconscious thought conscious? The ability of "my consciousness" to reflect about it.
>A related question is "What actions (in minute detail) are involved as I believe/think/feel I'm conscious?"
I think we can define a generalized version of what we call a "conscious thought" as a thought that can be "consciously reflected" about. With "conscious reflection" I mean using some representation of the thought as an input for another thought in such a way that the new thought, including the usage of the original thought, can be used as an input for another thought in the same way. The representation doesn't have to correspond particularly closely to the execution. (We remember to think in natural language, but maybe the thought process is just converted into natural language after the fact for the purpose of saving and reflecting.)
Computer programs can also be conscious according to this generalized definition if the process of executing it is saved in some way and can, depending on the circumstances, be used in some way, which is saved in the same way.
That different "ways" of using a thought as an input for another thought are possible means that there can be different consciousnesses in our brain. One of them (if there are multiple), which is the one responsible for our output, at least on a higher level, or does at least significantly inform the output, is what we call "my consciousness". The reason why we don't know whether "my consciousness" is responsible for the output, despite having the appearance, is that it may be mostly a "post-hoc rationalization engine" for decisions made on another level, possibly for other reasons. But it does at least inform the output, since thoughts of "my consciousness" in the past inform our later output. For example, if we are asked about our past thoughts, we talk about the ones that are remembered in "my consciousness". This is the thing that puts it in such a special position compared to other hypothetical consciousnesses, which don't inform our output in this way, and are therefore invisible to other people, and obviously also to "my consciousness".
(Answer since edit doesn't seem to work.)