Interfering with the ball is sufficient to be called for off-side - you need to be in active play, but you can do that when your team doesn't have possession.
What you are describing would make attacking the keeper pointless for example when they dropped the ball from hand for a kickoff, cause that would be offside which is not true.
Ball is rolling towards the goal - if the attacking player touches it, it will be offsides: A player in an offside position has entered active play by "interfering with play by playing or touching a ball passed or touched by a team-mate".
Goalkeeper makes a save, since the ball is rolling towards the goal. If the attacking player touches it, it will be offsides: A player in an offside position has entered active play by "challenging an opponent for the ball" or "gaining an advantage by playing the ball or interfering with an opponent when it has [...] been deliberately saved by any opponent".
The goalkeeper deliberately plays the ball. Immediately afterwards, the attacking player receives the ball played by the goalkeeper - this is not offsides, since "a player in an offside position receiving the ball from an opponent who deliberately plays the ball, including by deliberate handball, is not considered to have gained an advantage, unless it was a deliberate save by any opponent."
Sure, but that doesn't apply here. If you watch the video, the defender passes back to the goalkeeper. Backpasses by the defending team that are intercepted by the attacking player (even if they are in an offside position) are not considered offside, regardless of whether the attacker touches the ball first or the goalkeeper. From your link, this is the pertinent line:
"A player in an offside position at the moment the ball is played or touched by a team-mate is only penalised on becoming involved in active play by:
...
gaining an advantage by playing the ball or interfering with an opponent when it has:
- rebounded or been deflected off the goalpost, crossbar or an opponent
- been deliberately saved by any opponent
A player in an offside position receiving the ball from an opponent who deliberately plays the ball, including by deliberate handball, is not considered to have gained an advantage, unless it was a deliberate save by any opponent."
By your description Mandzukic's WC final goal should be offside https://www.youtube.com/watch?v=WzN-ahqULc4