I agree with Art, it could well be the switch and not the lock and associated lockout mechanisms such as the key removal cable or neutral (Park) safety switch to the starter solenoid. Once you get the key in the lock and are able to turn it, it's basically the ignition switch providing the detente positions and spring loading. Like Art says, something jammed inside the switch, like a broken/bent badly worn contact. It's basically sealed, so foreign matter shoudn't get in.

To the OP, definitely soft key material makes for a wear problem, but as you say, why both keys? I guess it's possible for something to prevent the key barrel from turning all the way in the ignition. The pins in the keyway aren't generally a wear problem, but a long pin could get bent from brute force or get a burr on it from abnormal wear. Just for jokes, if you haven't done it already, try throughly flushing the keyway and even around the barrel with spray WD-40 to see if that dislodges anything, followed by a lock lubricant.

Now I've had to replace two ignition switches in just the last year due to wear. Both ignition lock and ignition switch wear can be aggravated by using a heavy key set. Have you been using a heavy key fob or have a lot of keys on it?

If it is a worn ignition switch, as Art and I are guessing, then replacing the ignition switch section on the back is quite straightforward. 940/960 and 850 ignition switches are all the same. You just pull off the wire connector from the back and remove two slot head screws. Easier said than done though. In 940s, and likely worse in 960s, the two screws are extremely difficult to access from below. The upper screw is particularly daunting, working blind using assorted small bent screwdrivers, skinned knuckles, much profanity and a sore back.

Much, much simpler is to pull the instrument cluster and work from above where you can get your fingers in and easily see what you're doing with the aid of a flashlight and a hand mirror -have the kick panel off underneath for added light and to catch dropped screws. Pulling an instrument cluster the first time can be a bit daunting, but once you get to know what trim needs to be removed in order to get at the screws it's a lot easier. Practice makes perfect. In the 940s, and likely the 960s, there are slots in the metal bezel where you poke in a long and thin shafted flat blade screwdriver to pry the side clips inward so you can pull out one side then the other, Around the edge of the cluster you may need to slip in a few wide thin blades to separate it from impressions in the dash material. IMO, well worth spending the extra time to learn how to do it compared to struggling from below the dash. Additional cluster removal tips can be found in the FAQ, in posts both here and in other Volvo forums and elsewhere on-line, maybe even a Youtube video if you're really lucky. Start a separate thread if you need more help with removal.

Now, if you really think it has something to do with the key removal cable, then (this applies to 940s and is likely similar in 960s), remove the ash tray and holder and lift the metal cross bar off its pin on the right. Right in the middle below that is the cable with a hex-sided adjuster sleeve. You could always try pulling that back and forward a bit with bent needle nose pliers to see if that makes any difference in key travel. I recall the cable is loosest when in the Park position, but it may be the other way around. The end of that cable has a ball ferrel that sits in a spring loaded lever under the shifter surround.
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Dave -still with 940's, prev 740/240/140/120 You'd think I'd have learned by now