I own same corolla GLi 2009 and observed the problem in first start. I know ignition is result of Air+Fuel mixture in presence of spark. I just make a habit of cracking 3 seconds after switching ON (The full controls ON/ Or simply saying double switch) so that fuel starting from oil tank may reach to injectors. Or Simply cut-off first switch and again double switch (for second pulse of fuel). There is no rush in life. I know it will definitely resolves most of these problems.
Most of the problems get complicated when mechs don't use proper tools to check like pressure/flow meters before changing Oil Pump, Injectors, Battery.. WTH they idiots.. what is the thing they cant check .. like battery volts/amps Should be measured if Voltages are REALLY down??? rather looting the owners with selling costly spare parts instead of improving their own Skillset & just avoiding idiotic approaches.