The umpire is correct. This is actually call the "3rd Strike Rule", not the "Dropped 3rd Strike Rule". Many times, umpires will refer to it as the Uncaught 3rd Strike Rule.
It is real simple. If the ball must go directly from the pitcher's hand to the catcher's glove/hand and be caught for the 3rd strike in flight to cause the batter to be ruled out.
Absolutely correct MTR. It never ceases to amaze me that Coaches continually forget or don't know about this rule. They believe anything that is caught cleanly nullifies this rule.
Here's some dogpiling on the subject:
ASA rules during a travel ball tournament. I'm the PU.
I had a girl who's dugout was on the 3B side. With two strikes and no one on base, she swings at one in the dirt. It clearly one hops into catchers glove who holds on to it. I do not call her out, I just signal strike and verbalize Strike 3. She begins walking to her dugout, catcher returns the ball to pitcher. It was only the 2nd out, so as sometimes girls do, they begin to converge on the circle to congratulate their pitcher on the strikeout (14U). Before the batter reaches the dugout, her coach yells at her to run to first base. She was almost to the dugout, so, she drops her bat and begins running. (A few more steps and my job would have been much easier)
Mayhem ensues.
As she's running across the diamond from 3B side, the other coach now quickly yells for the F3 to go back to 1B with the ball. Batter is running across the diamond at this point with a bunch of girls in the middle scattering like bugs. F3 goes back to 1B without the ball which now requires a throw from P who had it. Batter is dodging girls and the throw beats the batter by about 4-5 feet.
BU awards an out.
Offensive coach immediately comes over and argues obstruction.
After a meeting together, we ultimately stick with the out call, it was close, but both of us believed there wasn't enough for OBS.
How's all that for some fun in the sun?