Can anyone provide more information on this? Are there any actual published or internal rules that are used to determine if the passenger is at fault or airline when it comes to missing a connection when the first leg arrived late due to flight delay?

1) If I exit the plane down the stairs, and outside they have some agent holding a sign with my next flight, can I ignore their van transfer to the terminal and take the slow bus instead?
2) If I do use the representative am I obligated to follow his instructions to run or walk fast?
3) If I'm sitting 1A do I have to get off the plane first or can I take my time and exit last?

The reason for these questions is not to take advantage of EU 261 and purposely miss the connection flight for the 600 Euros. It's to hold airlines accountable for the delays that they could have controlled for, and for flight itineraries sold with connection times they choose to publish.

The airline is doing its absolute best to try to get you quickly on the connection flight, I commend them, however it might not be in the passengers best interests:

First reason, some people don't like to be rushed, it's stressful, and would rather miss a flight and even stay in a hotel overnight then to run through airports. Passenger books flight with 3hr connection they might have wanted to relax, shower, shop, eat, inject insulin, and while they would accept flight delay circumstances that lead to a reduced connection time equal to the MCT, they don't want a rushed 20min connection.

Second reasons, Had a 30 min connection due to delay, I've made the connection flight but my checked baggage didn't. This is much more disruptive to my travel plans as it could take days or weeks to get your baggage, and with Lufthansa you don't get any useful help. I'd rather take the next flight knowing my baggage is onboard. EU 261 doesn't help with delayed baggage.