Okay, just to clarify.
There is no specific start up sequence. In many aircraft (Airbus twins specifically) the right then the left are started in sequence to enable the correct pressurisation of the braking hydraulics in the event of a problem during push back.
Originally, in my company, the 747 (200 series) was started 4-3-2-1. This could be modified however if you were starting on stand or had snow and ice under the wing during push back, workers close to the aircraft during start up etc…
The later 747’s (400 series with uprated APU, it’s always been difficult to cool a 747 in hot countries!) came with a much more powerful APU which enabled 2 engines to be started at a time. Some companies do outboard and then inboard, some vice versa. My company sticks with a standardised SOP which is right then left.
So, to answer the question fully, there is no over-riding correct start sequence for starting 4 engined aircraft. There are SOP’s and recommendations based upon environmental restrictions/cross fleet commonalities but no hard and fast system based sequence requirement.
(edited to add: You can indeed start both engines at the same time on a Twin, it’s SOP for our 777’s. The limiting factor is the APU pneumatic duct pressure. :D )