"But he also argues that it doesn’t even imply that it is probable that the sun will rise tomorrow. But given that the very word probably means different things to different people, what exactly is he arguing against?"
The definition of the word "probable" doesn't really matter... whatever it means, Hume's point is that just because the universe operated a certain way in the past... it doesn't simply logically follow that the universe will behave the same way in the future. Whatever the prediction is that people are making, it will have the same inductive problem.
"From a betting perspective, it seems obvious that one should bet literally as much money as they have on the prospect of the sun rising tomorrow. Is this also unjustified? Why isn’t the very idea of someone who would have bet on this literally winning every single time in history not a good rationale for this being justified?"
It begs the very question. You're making a prediction about the future based on past events, and so the inductive problem presents itself. We have a certain intuition that if the universe operates a certain way in the past, that gives us some information about the future. Hume's point is that we cannot make a logical deduction from past to future.
We need to inject some other principle like the future will resemble the past to make our justification. But we can't get the principle "the future resembles the past" from the mere fact that in the past, "the future resembled the past". So the question is... how do we get such a principle?