I doubt flipping the wires around will change this issue.
How sure are you about this replacement? I can't seem to find any information on the Millionspot H200S09 relay.
The screwdriver is a triangular tip, I made mine grinding down an old Phillips head screwdriver since I didn't have the actual screwdriver.
Click on graphic to enlarge.
In the schematic, both U1b and U1c are amplifying the current sense signal. U1c, the "overload" detector, has a gain of 148, but it is slowed down by the resistor and capacitor on its output. The time constant (RxC) is 2.6 seconds. Multiple short hits to this RC combination would charge it up until it trips the relay latch.
U1b has a gain of 37, which means it requires 4 times as much current, but it acts instantaneously for "dead short" situations.
All I could find was a photo of the bottom of a Millionspot H200S12 relay. It looked a lot like the Omron. I couldn't verify dimensions or pinout, but most of these Chinese relays are knock offs of the originals. I assumed that the H200S09 is a 9 volt relay.