There is definitely ambiguity to the term killscreen. Both interpretations are reasonable and there is no answer that is inherently correct. It is up to the community to choose the best definition based on the nature of the screen/the bug, the respect associated with a killscreen and under what circumstances it is deserved, and the objective of the game in general. With that said I think that the definition that should be chosen is that you have killscreened Donkey Kong if you have reached board 22-1, regardless of how you die on the board.
Firstly, the objective of Donkey Kong is to get as many points as possible and a definition should not be chosen that conflicts with this objective. Suppose, for example, a player has a 90% success rate at doing the back-jump over a barrel while grabbing the hammer. If the player chooses to go for the back-jump on the killscreen they get 800 points if they succeed or 200 points if they fail, an expected number of points of 0.9*800 + 0.1*200 = 740. If they choose not to go for the back-jump they get a guaranteed 700. It is mathematically the correct decision to go for the back-jump in this situation, and if they get unlucky and fall into the 10% case they and die, they still deserve all the prestige of reaching a killscreen. If you make the definition of killscreen dying from the actual timeout then a player in this situation is torn between making the decision that they know gives them the most expected number of points and the decision that gives them the recognition of a killscreen.
Also, there is nearly no challenge at all on the killscreen to survive long enough to trigger the screenkill. It's really board 21-6 that is the last of the challenges, after which the player should deserve the recognition of getting a killscreen. If a player really wanted to I think they could even just walk right and climb the broken ladder to get a nearly guaranteed death from screenkill. There is nothing special about doing that that warrants any more recognition than dying in any other way on the killscreen.
Some argue that a player hasn't reached the end of donkey kong unless they have died from timeout on the killscreen. This argument is implicitly associating surviving longer with making it further in the game, an association that makes no sense. Has a player who hides in the lower left corner until timeout really made it further than a player that climbs to the 3rd girder but dies from a barrel? Donkey Kong has never been a game that is purely about survival, it is a game about clearing boards so it would make more sense to consider the true end to be making it as close as possible to the end of the board rather than surviving as long as possible. I would argue that it's silly to make such distinctions and that all games that reach board 22-1 have reached the end of donkey kong.
Also, there's the argument that it's not a killscreen unless you've triggered the specific bug that is the cause of the impossibility of the killscreen. In response to that I would argue that you have triggered the bug as soon as the killscreen starts. The timer is actually set to 400 as soon as the screen starts, and beyond that point it behaves like a normal timer. In other words, dying from timeout on the killscreen is no different from dying from timeout on any other screen, it's the setting of the time at the start of the screen that is the actual bug. Anyone that reaches board 22-1 has triggered this bug.
