This is not for sure, this is what I assume, and shouldn't be taken as fact... But it should be pretty close (Just doing it because no one has replied yet...).
A client can be a client server, which then has their own clients, which are as you said the player...
So, let's go through the games (I'll get back with this later).
Sync. This can refer to animation syncing, packet syncing, etc. Like, you can't have someone seeing themselves jump, but to the other people, they see a player walking through the boxes they jump over. Also, you can't have people fake the packets or what not and let them go through the walls and what not. http://www.cocos2d-iphone.org/forum/topic/1615 A lot of aimbots hacks go off this same principle, and so it's actually useful for both sides =P.
As for Mafia 2, this could refer to the idea that, you don't want the game to lag... Like, CoD BO and Other CoD games after the year 2005 (I hate modern CoD if you can't tell), they just rape your connection to hell and back, and then cut your nuts off afterwards, but regardless, this is a great example of crap sync, another game is Homefront (Not a known game but w/e). So this is especially the actual developers fault... Or a big headed corp. not taking more than a year to release an actual quality game ahem CoD.
As for that game, I'd say it's because they're developers... For instance, in CoD2, it is VERY difficult to get custom models and especially animations in to the game... So it's the same principle most likely... I've never played Just Cause, but I'm assuming again... Not being in the actual game dev process, you're obviously losing some of the experience... Lol.
And yes. Everything networking wise has synch, whether it's games or files, or what not... Things have to be aligned to really work... (Again, assuming). Like, everything is a computer, watches, calculators, toasters, speakers... It's everywhere... And yes... If you look up the Zero To Hero guide, it comes with a Network Syncher... And I believe the Unity Networking tutorial also has one... It's very important... And really it returns a small value that has the XYZ of rotation, movement, scale, and that's usually it as the actual server does the math or what not for collision...
It's very complex, you should read up on it... No one here is an expert by any means I'd say... But I did my best and I'm probably wrong about some things, but this is a general gist.