Charm person/monster "If it fails the saving throw, it is charmed by you until the spell ends or until you or your companions do anything harmful to it. "
Charmed condition "A charmed creature can’t
Attack the charmer or target the charmer with harmful
Abilities or magical
Effects."
Dominate person/monster "which it does its best to obey."
Suggestion "it pursues the course of action you described to the best of its ability. "
If charmed vs charmed the target just can't hurt the charmers.
If charmed vs suggestion the target follows the suggestion unless it would be deliberately harmful to charmer.
If charmed vs dominate the target follows the domination as long as it does not attack the charmer. If ordered to attack the charmer the target "does its best to obey" then fails stopped by charm, having tried its best and failed the order is complete, "If the creature completes the order and doesn't receive further direction from you, it defends and preserves itself to the best of its ability." Which the dominator can continually do with the same effect.
If Dominate vs Suggestion the target "purses the course of action you described to the best of its ability. The suggested course of action can continue for the entire Duration." then:
A. Dominator
does not use their action on the following turn to "take control" and the telepathic command does not conflict the target does both. Example, Ordered to run away by suggestion, dominator says attack "that guy". If "that guy" is further away from the sugggestor than the target, the target will attempt to run away in the direction of "that guy" will take swing at "that guy" completing the command of the dominator then continuing with the suggestion to run away in that direction.
B. Dominator
uses the next turn action "to take total and precise control of the target. Until the end of your next turn, the creature takes
only the Actions you choose, and doesn't do anything that you don't allow it to do." Then Suggestion tries to "pursues the course of action you described to the best of its ability." but can't because dominate specifically denies it, effectively suspending but does not end suggestion as long as the dominator uses their action each turn to maintain "total and precise control" however, after 1 minute when dominate ends, if suggestion concentration is still maintained, the target will follow the suggestion again for the next 7 hours an 59 minutes unless interrupted.
C. Dominator
does not use their action on the following turn to "take control" and the telepathic command does conflict the target does both. Example, Ordered to run away by suggestion, dominator says attack "that guy". If "that guy" is closer to the suggestor than the target, then this is the only direct conflict. The target attempts to do both to the "best of of the targets ability" both continually failing and the target stands there holding its action until until something happens to change the situation. If "that guy" moves in range of the target, the target attacks completing the order and "If the creature completes the order and doesn't receive further direction from you, it defends and preserves itself to the best of its ability. " The dominator can issue the command again for free to hold him there but if he does not suggestion will cause the target to run away on his next turn. If he does and "that guy" is still there see A, if "that guy" moved repeat C.
That is how interpret it any way. i never seen anything that doesn't fall under these and only Dominate vs Suggestion C has conflict which "to the best of its ability" and "does its best to obey" results in a "if they can't they don't but they will keep trying".