We run combat according to the Advanced Melee rules, which are clear on this question.
- Initiating HTH is an action available for any figure moving 1/2 MA or less**. This includes engaged figures who can only shift.
- The move onto the defender's hex happens during the movement phase.
- As the attacker moves onto the defender hex, it drops any non-HTH weapons and shield in the hex it entered from.
- The defender then rolls to see if they repel the attack, get their dagger ready, etc.
- If the attack is repelled, the attacker return to the entry hex but without its weapon/shield, now laying there on the ground.
- If the attack is not repelled, the attacker and defender go down in the defender's hex.
- In any case, the attacker has used their action for the turn.
So to answer the OP's question, the attempt to initiate HTH is resolved during the movement phase (alternative 1).
** given the restrictions, ie attacker has higher MA OR defender backed against a wall, prone, or kneeling OR attacker enters from rear OR defender agrees