Okay, I got more details now, the pump was ordered from ebay, SKU ZPN-00922, description 12 volt SU style electric fuel pump 1-2 Psi Morris Minor
Ordered on the 12th of January this year, fitting date is unknown. The pump is New Zealand made.
The previous owner has fitted an inline fuel filter a few inches before the pump (vertical mounting). I can see the level in it being a bit low when it occurs. That's why I suspected that the pump might be mounted too high up the firewall, and could have trouble sucking the fuel up...
So I should remove the inline filter altogether and clean the pump's mesh and try my luck again?
I did remove the fuel cap, I did not notice a change of pressure, but yet I doubt that cap is sealed at all.