The umpire's motion isn't vague. It's a "safe" call. But by the time he makes it, the catcher has already started running away from him.
The important thing is that he never said "out", which makes everything that follows the catcher's fault. You can just assume you've made a play, you have to hear the words.
Yeah, there is a clear "safe" signal, but I don't know why and I had earlier stopped when the catcher's head starts to turn to the pitcher, and at that point, the ump hadn't yet made the motion. He did, though, earlier make the out signal. Which is also the signal for strike. The out and strike call should not be the same movement. Especially for this specific instance. It was third strike, the pitcher gets the K, but because the ball was dropped, the runner wasn't out (
7-umpire-signals-every-baseball-fan-needs-to-know).
Not sure why he made a safe call (his other arm is cut off from view, but it was moving into the safe call motion, so it looks like he did make a safe call). He was supposed to make a dropped ball call at that moment as the runner wasn't safe at first at that point.
"the signal used to indicate the third strike is by pointing with the right hand and arm out to the side using no voice."
But, yeah. As I already noted, the defense really fucked up. You never assume anything. You do what you need to do then if you want to celebrate, argue a call, whatever, you've already done what you needed to do. (that's also one of the reasons I hate watching a pitcher watch the ball rolling on the line when it's clearly not going to go foul, and by staring at the ball and not picking it up, the runners ar emoving, an potentially scoring; yeah, maybe picking up the ball when they pick it up will get the ump to call it foul, but that's a heck of a lot better than starring at the ball, not picking it up, runners score, and it then is called a fair ball - "A ball in the infield may start out fair and then roll foul. For this reason some defensive players may decide to let the ball roll foul if they think they can't get the batter out. They may also try to field the ball quickly and get the batter out before the ball can roll foul. Even if the ball goes back and forth between being fair and foul, it will not be ruled as fair and foul until it stops or a player touches it." - that's also a reason someone would intentionally not catch a foul ball, because once caught, the ball is now live, and if there are runners on base, they can now attempt to advance, and the person who caught the ball might not be in position to get the runner out. I have seen runners on base advance on foul ball outs, but I've never actually seen a player intentionally not catch a foul ball. Instead I see them stare at a ball rolling around the ground and continue staring at it instead of picking it up. The bloody ball might roll foul, yes, but the defender might also get the ball and throw out the runner before they get to first by picking up the ball. Infield foul is not foul/fair until ball stops rolling, defender graps the ball, etc. ).