In case you are not as interested in looking at a lot of mechanisms, I would direct you to just the snap motions ones at the bottom of the site. There are a few dual action versions which might be helpful for fixed "up" and "down" positions. The diagrams are not in numerical order but 7, 7B, 14, and 8 are interesting to me. Snap motion #8 is very simple which means it could be easy and inexpensive to build and less likely to fail. The blue element would be the flagpole with the forked end as the base and the other end holding the flag. The flag end could either just be extended to make the flagpole, or a pole could be attached to the end. If it is attached you could have the option of maybe pulling out a pin and being able to remove or wave the flag on its own. The total length of the blue element would be limited to the width of the backpack that holds it unless you add some type of telescoping solution. I see the white section being at the top, but still inside the backpack, to provide the greatest height and also concealment until activated.
When the flag is up, one manual pushbutton could push against the blue flagpole (at a point above and to the right of the red pivot point) to tip it over enough to get it off the flat portion of the base and onto the curved section. Then the spring would automatically pull and snap the pole down. This should require very little force and very little movement of the button. When the flag is down, another manual pushbuttom (which could push down a lever) could push up the bottom of the purple plunger. I would then mount a new block or cam just above the red pivot point and to the right of the plunger's upward travel. When the blue element starts to rise, this block would force the blue unit to tilt vertically. Once it is high enough the button would be slowly released to allow the flat base of the flagpole to settle and rest on the white base.
In addition to the total length of the blue element there is another "critical" dimesion. The distance the "up" pushbutton needs to travel is slightly more than the distance between the flat base of the blue element and the red pivot point. This does not need to be a great distance. The fork does not need to much greater than the diameter of the flagpole. This also allows the purple plunger to be much shorter than the example.
This orientation allow the flag to snap down but rise more slowly. You might want to rotate the design so the flag snaps up and falls slowly as that would be more dramatic. My initial description was made to match the animation. Once that is understood you could make changes. I hope this makes sense or more important gives you some ideas for your project. Good luck.