The
ejector is held by the sideplate pin in the left side of frame.
The
extractor is pinned in place on the right side of the slide. The pin
isn't staked.
Don't sweat mixing up the names, as I still do it upon occasion. Too many years of using both terms too many times.
The extractor pin is pressed into the slide with a HEAVY press, and it typically doesn't move easily. (Understatement alert)
Armorers typically use a "starter" pin punch and 4 oz ball peen hammer to break the pin loose (removed
downward, installed
upward), and then a 1/16" straight pin punch to finish driving the pin downward.
Pressure must be kept against the side of the extractor as the pin is driven down & out, as the extractor spring is often strong enough to
shove the extractor outward (once the pin clears it) and bend the thin shaft of the 1/16" pin punch (which is how new "starter" punches are made, from punches from which thin shafts have been broken off at their base

).
A thin tipped nail-set punch can be used as a starter punch, and it often takes a disturbing number of whacks with the ball peen hammer to break a pin loose, especially on older guns (I think I've heard the expression 'beating on it like the wrath of god' bandied about a time or two).
The tapered tip of the starter punch shouldn't be driven down into the pin hole too far, though, to avoid wallowing out the top of the hole. Hence, switching to the thin, straight pin punch once the pin is freed up to start moving.
Reinstalling the extractor and its spring & pin is almost just as much fun.
Aligning the holes in the slide (above and below the extractor recess) with the extractor body hole ... while keeping the extractor spring properly aligned in its spring hole, under the extractor tail ... and then firmly tapping the pin UPWARD into the slide, using a small mallet or hammer to get the pin inside the slide far enough to reach the "upper" pin hole, above the extractor (meaning driving the pin downward, when the slide is sitting
upside down on the bench), can take some patience.
I use a wide tipped Starrett flat pin punch to finish seating the pin into the bottom of the slide, to make sure the bottom of the pin is totally flush and doesn't protrude at all below the hole (where it might dig into, and gouge, the top of the aluminum frame).
Armorers used to be giving very fine and sharp needle reamers to clean up the slide pin holes
from the bottom, to remove any machining burrs remaining in the slide holes. Better machining apparently negated the need for that, as I've not seen them even offered in the tool kits in recent years. Care had to be taken not to get carried away and ream out the holes, though, as that's just another way to ruin a slide. (Maybe another reason they aren't provided anymore.

)
It's a good idea to check and make sure no steel burrs have been shoved outward/downward from the extractor pin hole, where they could drag on the top of the aluminum frame rail. Armorers used to be given a thin offset file to run along inside the slide rail, to remove any burrs created when removing an extractor pin, but the slightly upward circular raised area around the bottom of the extractor pins in newer production slides seems to help prevent that problem.
It's common to see a space above the extractor pin inside the slide's hole when viewed from the top, but the bottom of the pin should be flat with the bottom of the slide hole.
Why not just have the gunsmith remove the extractor when prepping for whatever it is you're planning to have done? Better than getting frustrated and risking ruining your gun.