I have a Mattituck (Lycoming clone) IO-360.
Their recommendation is similar to that shown above for other Lycomings, and it works pretty well for me.
Some aircraft are fitted with a purge valve, which, when opened, allows fuel to be pumped through the entire system as far as the distribution block and returned to the fuel tank. In principle this allows you to start with fresh cool fuel in the lines, but in practice not many people use it as it adds a potential critical failure mode into the system if the valve fails.
Here are Mattituck's recommendations:
How do I start my fuel Injected Lycoming style TMX IO-360?
Procedure for a cold engine:
(1) Set propeller governor control in "Full RPM" position (where
applicable).
(2) Turn fuel valve "On".
(3) Open throttle wide open, move mixture control to "Full Rich"
turn boost pump on, approximately 3 to 5 seconds, turn boost pump off,
then return throttle to "Closed" and return mixture control to
"Idle Cut-Off".
(4) Open throttle 1/4 to 1/2 of travel. Keep you hand on the throttle during the staring
process to make movement toward the idle position after the engine has started an easy
immediate thing to be able to do.
(4) Set magneto selector switch (consult airframe manufacturer's
handbook for correct position).
(5) Engage starter.
(6) Engine starts.
(7) Retard throttle towards idle position.
(8) Move mixture control slowly and smoothly to "Full Rich".
(9) Check oil pressure gage. If minimum oil pressure is not indicated
within thirty seconds, stop engine and determine trouble.
Procedure for a hot engine that was shut down with in a few minutes ago:
(1) Set propeller governor control in "Full RPM" position (where
applicable).
(2) Turn fuel valve "On".
(3) Open throttle wide open, move mixture control to "Full Rich"
return throttle to "Closed" and return mixture control to
"Idle Cut-Off".
(4) Open throttle 1/4 to 1/2 of travel. Keep you hand on the throttle during the staring
process to make movement toward the idle position after the engine has started an easy
immediate thing to be able to do.
(4) Set magneto selector switch (consult airframe manufacturer's
handbook for correct position).
(5) Engage starter.
(6) Engine starts.
(7) Retard throttle towards idle position.
(8) Move mixture control slowly and smoothly to "Full Rich".
(9) Check oil pressure gage. If minimum oil pressure is not indicated
within thirty seconds, stop engine and determine trouble.
Procedure for a hot engine that was shut down more than a couple of minutes ago:
(1) Set propeller governor control in "Full RPM" position (where
applicable).
(2) Turn fuel valve "On".
(3) Open throttle wide open, move mixture control to "Full Rich"
turn boost pump on, approximately 1 second or less, turn boost pump off,
then return throttle to "Closed" and return mixture control to
"Idle Cut-Off".
(4) Open throttle 1/4 to 1/2 of travel. Keep you hand on the throttle during the staring
process to make movement toward the idle position after the engine has started an easy
immediate thing to be able to do.
(4) Set magneto selector switch (consult airframe manufacturer's
handbook for correct position).
(5) Engage starter.
(6) Engine starts.
(7) Retard throttle towards idle position.
(8) Move mixture control slowly and smoothly to "Full Rich".
(9) Check oil pressure gage. If minimum oil pressure is not indicated
within thirty seconds, stop engine and determine trouble.