Hi Child of the island!
As to how I understand the first passage, the thought “I am” does not occur because of “depending” on something. It happens because you take something to be yourself. See how a similar idiom is used in English (‘take to be me/mine’)? The idea is not that the identification needs something (depends on something), but that there is this ‘thing’ (for example the body) “out there”, doing its own thing, as it were. The mind goes and identify with it, taking it as “me” or “mine”. The identification is always a sort of afterthought which happens after a thing has come into consciousness. For example, first there is seeing through the eyes, then the mind registers the sight, and only then it adds the layer of “I see”.
The mirror simile I find difficult to translate properly into English. The specific verb form upādāya (called a ‘gerund’) is often used idiomatically, often not directly translatable into English. I suspect some idiom here too, but the exact idea seems lost. There is little information in the passage to go by, and I find all the translations I know to be a bit clumsy. But ‘dependent on’ also isn’t the solution, in my opinion.
Then the other two passages. First of all, thanks for pointing out something I glossed over in my initial post, which is the meaning of upadana of ‘fuel’. I treated that very briefly, hoping to keep the post as simple and to the point as possible. But I regret this now, as I’m convinced ‘fuel’ is the main meaning of upadana in dependent origination. It seems that whenever the noun (upadana) is used, it almost always means ‘fuel’, while only the verb forms (upadiyati, upadaya, etc) generally mean ‘taking up’. I may add some specifics on this later. (Unfortunately, it seems I can no longer edit my first post. If any moderator reads this and can enable that for me somehow, that would be great.)
So in these passages, where we have the noun upadana, I also see no reason to translate it as ‘dependence’. This is simply not the meaning of the word. In both passages it means ‘fuel’. In SN44.9 it is literal fuel, the kind that a fire burns, which is “taken up” or used up by fire. As I pointed out before, you can compare it to the English noun ‘uptake’ which means absorbing nutriment (“fuel”) by an orgamism. In SN22.82 it is metaphorical fuel: the fuel for rebirth, being desire.
And as to your last comment, I don’t know any place where ādāna would mean ‘binding’. If such a passage would exist, it would be a rare idiom–one which I think shouldn’t influence the interpretation of other passages, because the general meaning of ādāna is already clear enough.
Hope this helps!