Yes, your understanding is correct.
"He wanted his dad to trust him."
In this sentence, "him" refers to the subject of the sentence, which is the person wanting the trust. So, "him" refers to the person (not the dad). If Tom is the subject, this would mean Tom wanted his dad to trust Tom.
"Tom wanted his dad to trust himself."
Here, "himself" refers to Tom (the subject of the sentence). Reflexive pronouns like "himself" refer back to the subject, so in this case, it indicates that Tom wanted his dad to trust Tom, not the dad.
So, in both cases, Tom is the one seeking trust, but the use of "him" vs. "himself" changes how it's structured grammatically.