I agree with you.
Today it's raining. I have to stay at home and I __2__ study hard because I __3__ find a good job in the future.
I think that questions such as the one above are pointless, because there is an argument for both 'must' and 'have to' in both gaps.
A further complication with these two verbs is that many Americans appear not to use 'must', so the difference felt by some native speakers is unreal for others.
For many speakers of BrE, 'must' suggests an obligation imposed by the speaker, 'have to' an obligation imposed by somebody else:
Teacher: You must do this homework tonight. (I, the teacher, impose the obligation.
Mother: You have to do this homework tonight. (The teacher has imposed this obligation, not I, the mother).
Even this difference is not always observed when the subject is 'I'. In an utterance such as "Sorry, I have to go now", the speaker may use 'have to' when there is no external obligation, to suggest a polite unwillingness to leave.
Some people appear to feel that 'must' is somehow stronger than 'have to'; however I have heard others claim that 'have to is stronger than 'must', so that's no help.