Note: Assuming that the game wants you to exit through the door at the end of the Right path for the achievement to pop, it's recommended to probably take the Left Path first to pick up all the goodies. Alternatively, there is a typewriter just outside of both of the paths...if you save before going for the achievement and letting it pop, you can simply reload in order to score a kill on the Gigante afterwards. Then you can take the left path if you choose, it's all according to preference.
I can confirm that the first El Gigante fight is near the middle of 2-1. After Leon passes out in the Cabin upon defeating Del Lago, you'll make your way back to the village coming across the first El Gigante of the game in an area you visited previously near the end of 1-3. This fight offers no rocks aside from it technically being in a Quarry, so killing this Gigante has little to no relation to the achievement:
A Rock and a Hard Place 50
Outmaneuver the rampaging beast, El Gigante.
After 2-2 (in which you and Luis defend the cabin from a Ganado invasion), you'll start up 2-3 and be outside of the cabin. There is a lever that can be pulled to open one of two paths leading to a large number of Ganados plus the Bella Sisters (female equivalent to Dr. Salvador, basically two chainsaw wielding maniacs) or to a large open area, seemingly empty only to reveal a El Gigante chasing after you. Regardless, both paths have a key somewhere within the level (located within a building on the Right path, the El Gigante path...and on one of the Bella Sisters, for the Left path).
Basically the achievement describes "outmanuevering" the Gigante, it doesn't say anything about killing him. In all honesty it's possible to get every important item (including treasures) in this area without killing him, you just have to be very fast and know where things are (meaning don't chase after a green herb in a hut if you don't need it). But the focus of the achievement appears to be outmanuevering him which is tricky to do if you don't slow him down with the boulder above.
So basically for this achievement, I assume what needs to be done is focus on the very first door you come across, blowing off the chains (a shutgun or explosive such as a grenade works best, knife takes several swings which you don't have time for)...then turn around and search for the boulder above.
You will see a wood support holding the bolder in place, that the red dot from your aiming sight will land on...you'll want to shoot this (not yet). There are two ways to slow down the Gigante with the boulder, the first (and most efficent way) is to wait until he's JUST before where the boulder is so that when you shoot the support, it will land ontop of its head. This will buy you anywhere from 10-20 seconds to turn around and shoot the chain off the next door (from memory, I can't remember if there's a second door here or not x.x) The other method is to shoot the support, causing the boulder to land in front of the Gigante...it will take maybe less than 5 seconds for it to stop, wind up its fist and destroy the boulder...giving you only a short amount of time to continue on your way.
Regardless, clearly the best option is using the boulder to hit the Gigante. After busting open the door, you'll want to run into the cabins searching for the Old key (along with picking up extra items along the way). When you've found it, if enough time persists you can search above this area for a bucket hanging overhead. Shoot it to reveal a treasure that you can sell for some Pesetas later. If however the Gigante is on your tail, it's faster to now just use one of your Flash Grenades on it. This will cause it to be stunned for about 10 seconds, giving you more than enough time to make it to the end of the area (and shoot the chains off another door if there is one). Simply open the door at the end and the achievement should pop (if the theory persists that you don't KILL the gigante).
EXTRA NOTE: If for some reason killing this Gigante also triggers the achievement to pop, there are several easy methods to insure a quick defeat:
1.) Use the boulder, then fire on the Gigante until the Plagas is revealed. You can then either shoot it or climb on it's back, knife swinging (your choice). Rinse and repeat til death. Flash grenades work wonders here too, if you have them USE THEM.
2.) If you have enough Pesetas on hand, buy up the rocket launcher from the merchant. As soon as the beast appears, one rocket should be enough to kill it. If for some reason it doesn't die, use the boulder to finish it off and/or blast away at the plagas on it's back for a quick kill.
Special Note for Professional:
Let the Gigante pick up Ashley in her Suit of Armor, while it's busy trying to kill her (which it can't) you can either fire a rocket at it or shoot it a lot. Your preference, eitherway using her as a lure is beneficial enough to you in order to make the fight easier.
Not recommending to stick all of this in the guide ;>__> considering how lengthy it is. But I'm positive that your description for it is right and you DON'T kill the second Gigante, rather you escape it (possibly having to use the boulder for the achievement as well).