Simple...easy...done...
Power an Arduino directly from that 10-14.5v from the pump circuit.
Arduino powers on, waits a few seconds just in case it's a 'spike', increments a counter by one, saves that number in EEPROM, spit that number out the serial lines say once per second.
Bilge pump shuts off, Arduino shuts off (and that's why it waits a couple seconds before incrementing that counter in case there's a 'spike' when the pump shuts off).
Hook up a PC to read the number...but you don't want to do that.
So, hook up a simple serial backpack or 7 segment LED, whatever you want, literally thousands of sketches already out there for doing this. When it's got power, it displays the number. But, it'll be one count too high when you get to reading it because you have to power it on to see it, so subtract one from that number. Have another push button set up to reset that number to zero or short a couple pins together on the Arduino itself.
Whatever works. Not that difficult.