Where am I wrong ?

Based on the original question, there was no mention of a cat or bad
running let alone mixtures inside the cylinder next to the spark plug.

Most cars lambda is measured after the engine and before the cat.
At that measurement point NO petrol car will run at lambda 2.0

As you have stated, only TRUE gas analysers can measure lambda post cat
and these are never seen in your local garage or tuning house - I agree.