Actually, there's at least one more category neither single nor double action, not sure what it's called
Example, striker fired, like a Glock
A Glock and many striker fired guns are not carried fully cocked , yet the trigger pulling does not fully cock them.
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Mainly what you should be concerned with these days, rather than DA/SA (in my opinion) is "trigger pull"
Example" I have a Double Action Taurus revolver with a heavy trigger pull to cock it ... that I can't hit a barn with firing double action (pulling trigger cocks, then fires)
Same gun if I cock it, has a light trigger pull to fire it and it's very accurate.
This is why guns like 1911s, Glocks etc have various triggers you can get.
Getting a double action with a hammer and cocking it, leads a lawyer suing you to maybe claim it accidentally went off with the light trigger (Negligent homicide instead of firing on purpose in self defense)
Many guns are DA/sa meaning if you do not carry cocked and locked (cocked with safety on) they''ll fire first shot DA (cocks, then fires, but subsequent shots SA (slide cocked it after firing the first shot) so your first shot and second shot have different trigger pull weights.
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DAO = Double Action Only, such as many semi-auto/hammerless revolvers have a heavy trigger pull every shot, the hammer decocks after each shot, ... every shot cocks ... then fires (hammer may not be visible, many recess the hammer)
But, just because you do not see a hammer, it does not mean it is DAO nor double action nor DA/SA ... it only means you don't see a hammer, it might be striker fired or hammerless.
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Seriously, rather than worry about the mechanics/ inertia firing pin whatever, try several and concern yourself with if you like the trigger and recoil and can hit the 10-ring ... or the barn ...or not
I look at, how fast and accurate is my SECOND shot ( can I be on target FAST with it?) I don't count on a bad guy watching enough movies to know he's supposed to fall down after just one tiny hole is made. Can I re-acquire a sight picture fast and be on target fast and accurate the second/subsequent shots?
That's the only disadvantage to a
DA/SA first shot Double action/cocks and fires-second shot is already cocked by slide movement from first shot ...
second trigger pull weight different than the first, that I notice, so guns like a 1911/P238/Taurus PT-92 etc I carry cocked and locked (cocked with safety on)
If I recall, HK P7 uses a lever under the front of the grip to cock, and trigger to fire with consistent trigger pull weight from shot to shot, it's a great gun if you are rich (You probably won't find one for sale)
Again. don't worry about DA/SA engineering/design etc so much right now, try guns, get what works for you.
Study features, study procedures how to clear jams etc tap/rack/bang, study safety ... later you can learn engineering
BTW
TAURUS PT709
Features
FIrst Shot is SA (good trigger pull) subsequent shots are too really I believe,
but they call it a SA/DA (not DA/SA) because if a round does not fire, you can pull trigger again on the same round (if it had a hard primer/light hammer strike etc) and try that same round again (It recocks and fires with 1 pull) THEN if the "second chance" still does not work, you can treat it like any other gun, clear the dud round and insert new by racking the slide once.
"Double strike capable" ... I like that feature. It's a main reason I got that model.