***What's the 100% correct way to bleed the cooling system after repair?***
The correct answer is "Till there is no air in the coolant system"
If any of you have ever taken your car to a professional shop for cooling system maintenance does the mechanic tell you to come back the next day so he can top off your coolant? Think about it.
I was reading this thread for fun and I have to laugh at some of the responses above - and since it is like an oil thread I was gonna just avoid posting till I read OnTheFence response above - especially when he says 3 times not to start the car. You got this from where - the BMW manual - a forum - certainly not from decades of bleeding various cooling systems on a wide variety of engines.
You posted a set of instructions
here a year ago where you asked if an e46 should be bled like an e36. The ones you posted above are similar and I didn't say anything then. I will remark on two of them now though:
Keep pouring until the coolant is bubble-free (this may take a minute or two).
You will spill a lot of coolant.
You will spill coolant with the engine off as there is nothing (waterpump) pulling in the coolant and the bleed hole in the t-stat is only a pin hole - so the front end of the cooling system is full but you're trying to pour more in and only a minute amount can trickle through the t-stat bleed hole. Amateurs try to pour the coolant in quickly on a closed coolant system and it spills or overflows like a frothy beer in a mug. Professionals don't spill coolant on the floor since the engine is running the ET will be empty if the system is low and as you pour the coolant in the waterpump pulls it in, pushes it into the coolant channels and displaces trapped air which cycles around the system and purges out the ET. The other worrying thing is the air bubbles you are seeing are only at the ET - all the coolant that just went in carried air bubbles you are not seeing to the entire cooling system and that still needs purged and burped and that can't happen till it's circulating.
The other instruction I think is ridiculous is this one:
Check frequently over the next few days and top off as necessary.
If you'd bled with the engine running t-stat open, waterpump running - you wouldn't have to check several times over the next few days as you've removed the air via purging and burping the first time - in the professional world who has time for checking several times over a period of days? - is the customer going to bring their car back 4 days straight - you gotta be kidding - do it once - do it right - here's your bill - goodbye.
Your task 2 above in this thread says "start car" OK so the OP drives around and guess what - if there was an air pocket he'll be topping it off tomorrow (hopefully) when the engine is cold - maybe a third day as difficult trapped air in the spiral heater core get purged out now the waterpump is pushing coolant instead of air bubbles - I'm not attacking you individually and I realise that's how BMW basically phrased their instructions for the amateur home mechanic - but that was to make it safer - for them - from lawsuits. The BMW instructions are about liability mitigation. Not sure why everyone on here pays attention to them for bleeding - probably one of the reasons this forum has likely more threads about poor bleeding & issues than every other auto forum in the world combined.
This engine is not any different than any other pressurised cooling system in any other car. It has nooks and crannies and spots that will trap air. If you've lost or drained coolant it has been replaced by air. The coolant you pour in will contain air bubbles - these need purged and burped. Purging only occurs when the coolant is circulating - burping occurs when the trapped air behind the t-stat is allowed to escape through the bleed screw or ET cap.
Not gonna write a how to on bleeding a bloody engine - but if you have questions - do a Google Search and read how an engine is bled with it running - don't pay attention on this forum for this aspect of e46 maintenance. - pros do it with the engine running Doing it cold with nothing running to purge the air just means you hope that after your next drive the air has been moved to the ET where you can just top it off per the safe amateur instructions - well you're really doing the same thing - just in two different stages and possibly without the same results in addition to possibly driving the car low on coolant before top off. Often the trapped air doesn't get moved even after driving since you've now closed the system and air compresses much easier than coolant with pressure and the result is 40,000 threads about e46 bleeding issues.
Edit: and ..... the reason many raise the front end when bleeding/topping off is so the escape route for air (ET hole or bleed screw) is higher than any other coolant component - meaning the air can be pushed through and as we all know air rises in liquid - can escape.