After the needle just moves (or after 5 secs or so if it doesn’t move).
The pump run is just to put some fuel into the pipework, and that fuel doesn’t disappear quickly, so if you wait a minute or two before starting the engine, that’s ok.
That’s fantastic all the scientific differents methods we use !
On my IO540, hot start is without priming, 1cm throtle, and 3cm mixture (from full lean).
Then I crank, and move the mixture slowly, otherwise, miss the startup.
I’m with boscomantico here: that’s just for me !
(and I struggled a lot too at the beginning ;-)
On my IO540, hot start is without priming, 1cm throtle, and 3cm mixture (from full lean).
That is probably the same as mine. There is no significant difference between 3cm mixture and full-rich.
As regards the throttle position, as stated above I am probably over-doing it and there is no reason for it to be bigger than the cold start position i.e. about 1cm.
So for all the different methods, most of them are probably similar.
I cannot fully agree to that summary ;-)
As per POH I start cranking the engine with throttle part open (POH 1/4) and mix cut out, and move mix forward only when the engine comes alive.
(only when it does not start with a prop turn, max two, do I start to move the mix in slowly…)
I start with an enriched mix only when the engine is cold….
In that case I also have the pump run until I see fuel flow like you ;-)
Good video for Lycoming IO360:
My experience on both the Aztec and PA31 is that when very hot (just long enough to fuel, or show your passports) it is more effective to be full throttle. There is plenty of time to pull the throttle back before the revs build to much, it rarely goes above about 1500rpm.
I also find that the tiniest squirt of prime (as quick as you can turn the pump on and off again) is helpful.
I have no explanation, but it does seem to go better that way.
That makes sense relative to what I have been doing for 15 years (IO540-C4).
I have to be fairly quick on the throttle pull-back though. It is also really embarrassing if you do it at the pumps with your feet off the brakes, and next to me is an FTO plane with three Arab students and an FI hoping to show them what a cowboy this private pilot is 
Personally on the PA31-350
Cold – throttle about half an inch, boost pumps off, fuel pump on, full mixture for about 6 seconds, mixture ICO, fuel pump off, engage starter, advance mixture when the engine catches.
Hot – throttle about half an inch, boost pump on, engage starter, advance mixture when the engine catches (this can take quite a few seconds.
On the DA62
Cold – engine master on, wait for glow light off, engage starter.
Hot – engine master on, engage starter. 
Starting a reciprocating fuel injected engine is an art!
Every individual engine is a little different.
But in general assume you’ve over-primed the engine. That’ll get you out of most trouble.
Try not priming, or only a very short (swift up-down) movement of the mixture.
One thing that has baffled me in French aeroclubs (or is it just mine) is their insistence on priming carburetted engines with the throttle lever instead of the primer. That is not according to the POH, it is a fire risk, it is less effective and harder on the engine, so I don’t understand why anyone would insist on doing it that way?
Easy. That is mainly tradition as many older french aircraft don’t have primer pumps – of the planes I have flown there is the D11, D120, DR1050, DR250, DR400, MS880, MS883, MS893 … just to name a few. The BO208 hasn’t a primer, too. In cold weather I (if possible) prime with the throttle with ignition off and just add the spark after a few blades.
