Metal Slug 3

In Mission 2, the president is underneath a path leading up and is tied up and you really can't miss it. Just crouch under the walkway and use your knife to cut his ropes.

Sometimes the president can be missed in Mission 2 if you are moving to fast as you can't backtrack because the screen wont move back for you, so just to be safe take it easy at the beginning of the level before you reach the hill.

Then free the president and you will be rewarded with 10G and a cool little Cloud powerup that is pretty useless to be honest.

(Credit to bipolardragon for the video)

Also in the second video which is a perfect run on Mission 5, you can save the President again/or for the first time. Keep in mind in order to save this President you need to first save Hyakutaro Ichimonji (0.11 in the second video), he provides decent support during this section of the level with his fireball and roundhouse kick abilities. Hyak' is also essential for this part as his fireball attack is needed to free the President from the top of the screen.

Follow the second video from 1.52. This shows you Hyak' shooting Fireballs at the top of the screen, which results in the President being freed.

(Credit to ex76823 for the second video.)