Video nr 1
The attacker is challenging the opponents for the ball, but before that the defender makes a deliberate play. When a defender makes a deliberate play "to save a goal from being scored" then it must be considered as a deliberate save and the attacker receiving the ball must be called offside.
Deliberate save by a defender
Video nr 2
In video nr 2 the assistant does not have to figure out whether this is a deliberate play or not because it is also a shot on goal and therefore it is a deliberate save, so it is offside.
Videos nr 3 & 4
The next 2 videos are examples of the most common deliberate saves by a defender.
Deliberate save by a goalkeeper
Video nr 5
The quality of the save is not important. When it is "an attempt to save a goal from being scored" then even a deliberate play must be evaluated as a deliberate save.
Videos nr 6 & 7
The next 2 examples are correctly called OFFSIDE.