Who gets the Commander if it dies or is exiled with a coin counter from Athreos, Shroud-Veiled?

The commander's owner always gets to choose whether the commander remains in the graveyard to get returned, or whether it goes to (and remains in) the command zone. The only difference between the old and current rules is that under the old rules, Athreos didn't even trigger if the commander's owner choose to move it to the command zone.

Previously, moving a commander to the command zone was handled exclusively as a replacement effect. A commander that left the battlefield for any reason wouldn't even hit the zone it was going to, but straight to the command zone, if its owner chose to. In that case, Athreos wouldn't even trigger, let alone return the commander under your control.

903.9. If a commander would be exiled from anywhere or put into its owner’s hand, graveyard, or library from anywhere, its owner may put it into the command zone instead. This replacement effect may apply more than once to the same event.

614.6. If an event is replaced, it never happens. [..]

Under the current rules, a commander moving to the hand/library from anywhere is still handled with a replacement effect:

903.9b If a commander would be put into its owner’s hand or library from anywhere, its owner may put it into the command zone instead. [..]

However, a commander can now move to the graveyard or exile from anywhere and thus trigger all applicable triggers. In that case, its owner gets a choice to move it to the command zone, before anthing on the stack can resolve, as a state-based action:

903.9a If a commander is in a graveyard or in exile and that card was put into that zone since the last time state-based actions were checked, its owner may put it into the command zone. This is a state-based action. See rule 704.

That means that under the current rules, Athreos would trigger on the commander's death, but if its owner chooses to move it to the command zone, Athreos' ability will not do anything on resolution because the card it referenced is no longer in the appropriate zone.

The commander's owner gets to choose.

• If they choose to put it into their command zone, it stays there. You don't get to reanimate it under your control.
• If they choose to leave it in the graveyard for some reason, you do reanimate it under your control.

You get the same outcome whether you're using the old rules or the new rules. State-based actions occur before players get priority, and before triggers go on the stack. The replacement itself would occur even sooner, at the moment the zone change would happen—if the command zone replacement effect was applied, the commander wouldn't even be considered to have died!

Either way, the option to move their commander into their command zone from the graveyard therefore happens before Athreos's triggered ability even goes on the stack.

Athreos's triggered ability will then go on the stack if it saw the creature die. (This happens under the new rules either way, or the old rules if the opponent opted to let their commander die.) If the player moved their commander out of the graveyard into their command zone via the state-based action, Athreos won't be able to find the commander anymore and won't reanimate them. From Athreos's rulings:

If a creature with a coin counter on it is put into a graveyard or exile but leaves that zone before Athreos’s last ability resolves, that card stays in its new zone, even if that zone is also a graveyard or exile. You don’t return it to the battlefield. (2020-01-24)

If however their commander is still in the graveyard because it wasn't moved, it'll be put onto the battlefield under your control as part of Athreos's ability resolving.