I think you are already on the right track with the crank sensor , the symptoms sound correct ....ok at cold start and trouble at warm start .
You say fitting the new sensor made it behave great for a period of time , the trouble we have at work with after market sensors I have lost count of the number of times..... crank , cam , air mass meters , egrs , temp sensors in the after market have caused issues and confused the diagnostic pathway testing .
I would be tempted to fit a genuine crank sensor and try again , the vehicle parts market is flooded with Chinese copy ting tong parts ......I would also say Intermotor are not the quality company that they used to be back in the day ....it's all about profit now and not quality sadly .
Here's how a ac generation crank sensor looks on a scope
Draw a horizontal line at the 0 volts through the trace , you have plus volts and minus volts because it's an ac generation sensor ....as a rule whilst cranking the ecm wants to see around 3 volts peak to peak signal amplitude , the change in the trace so far through is the missing tooth on the flywheel reluctor ring or front pulley toothed ring whatever the application , that missing tooth is how the ecm picks up the top dead reference .
So when crank sensor signal and cam sensor signal are seen to be in synchronisation when cranking this is when ecm triggers injection begin pulse.
What happens with the hot start is the heat soak affects signal amplitude from the sensor , in some cases jump starting it just gives the starter extra speed so a larger amplitude of signal is generated and away it goes ....or just let it cool off and it re starts also .
Of course it may be something else ......lol !