It could be argued that the second reference is correct, as it says Armageddon will be the final war "against Satan's world". Once the 1,000 year reign begins, the world will no longer be Satan's - he will no longer be the predominant force and his earthly organisations and structures will have been destroyed - so once he is released at the end, he will be in the minority rather than being the 'ruler' of that world.
However, the first reference would still be wrong in simply stating “these events will lead directly to the final war of Armageddon" without qualifying that.
What's also interesting is that that Rev 20 scripture still refers to "nations" - will mankind still be split into national groups, or is that a figure of speech, like "people of all nations", simply meaning from all backgrounds? Or will it mean that those near the end of the 1,000 year reign who choose not to live by God's rule will split off into independent 'nations' elsewhere on the earth? The verse also implies a physical location (or several) for the enemies of God ("they advanced and encircled...") and also a single physical location for the followers of God ("... the camp of the holy ones and the beloved city").