mw13 – I agree your first example is apikorsus, because it states that Hashem’s koach is limited, as though Hashem were incapable of giving parnoso without my help ch”v.
But I don’t agree with your “Ditto”, it isn’t a ditto at all, I would have thought its completely different. If someone believes Hashem COULD do it, but thinks that in the system Hashem chose to set up their hatzlocho does come from their hishtadlus, to the best of my knowledge this is not apikorsus, but is a lack of bitochon.
Your brothers teacher also, if he meant that Hashem CAN’T stop global warming then he is an apikores as well as stupid (nebach and an apikores, to misquote). But if he said that Hashem wants us to try to look after the world, and set up the world in such a way that problems can develop if we are poisheia, then this isn’t apikorsus. (For all I know this may be right, Chazal say (albeit about something different) “תן דעתך שלא תקלקל ותחריב את עולמי”, there is a potential for קלקל which is in the hands of people, it is possible that it is the same here, you would presumably agree that one shouldn’t pour petrol into the ocean etc as it damages the world. But lets not hijack this thread).
The term shituf as far as I am aware generally refers to belief in two ultimate koichos. It is conceivable that R’ Dessler or other Baalei Mussar borrow this loshon to describe bitochon in two sources, even though it really means emuna in two sources i.e. accepting them as a god. Thinking that you play a part in your success obviously doesn’t mean you accept yourself as a god.
It is true that people would have stronger bitochon if they had perfect emuna. However with every aveiro one could argue that if the persons emuna was stronger they wouldn’t have done it, that doesn’t mean they’re an apikorus. As long as they know intellectually and believe in these ideas, not having fully emotionally internalised them is a chisoron, not apikorsus.
[I have often wondered what the source of bitochon is. What parts are mideorayso, what parts perhaps miderabonon or minhag, which parts a good middoh. As most of the seforim which discuss bitochon are mussar seforim it can be hard to tell. I don’t think the Rambam brings a chiyuv of bitochon in Maddo or in Sefer Hamitzvos as one of the mitzvos min haTorah, indeed this may be part of the Ramban’s point in shikchas haseyin no. 8 “Tomim tihyeh”. However even the Ramban doesn’t include in that mitzva everything which is normally included in the term bitochon. Which is what leads me to wonder whether some parts of bitochon are a middo.]