Haven't been able to find any definitive information about this.
903.3. Each deck has a legendary creature card designated as its commander. This designation is not a characteristic of the object represented by the card; rather, it is an attribute of the card itself. The card retains this designation even when it changes zones.
This states clearly that commander-ness is a meta form of information, immutable to any form of play. However, it doesn't say this information is absolutely public.
I think this question falls into a black hole of oversight.
Commander-ness being an attribute is unique in the rulebook. It's a property of the card, but stops there. I think it's assumed to be public at all times, but it's not stated implicitly or explicitly anywhere. This feels wrong in a game where such clearances are very important.
- If you cast your commander face down, is it identifiable as a commander spell on the stack?
- If your commander is shuffled into your library, is it's position always known?
This might fall under Free Information / Derived Information under the tournament rules.
Free information is information to which all players are entitled access without contamination or omissions made by their opponents. If a player is ever unable or unwilling to provide free information to an opponent that has requested it, they should call a judge and explain the situation.
Derived information is information to which all players are entitled access, but opponents are not obliged to assist in determining and may require some skill or calculation to determine.
A simple amendment that commander-ness is defined also as free information / derived information might solve this issue.
I'm aware there is another question about this on the site. However I'm not satisfied with it's answer as it relies on information that is no longer available (EDH Rules Committee no longer has a forum that I know of), and the comments below that answer make some good counter-points.
So to reiterate, is a commander card always marked across all zones? Are there exceptions, and how are they defined in the CR?