I know the rule for playing scales using the thumb on the white keys - and not running from a white key played e.g. by the middle finger under leading the thumb to a black key - let's say you play the B major scale then it is clear that the thumb is on B and E and not (1,2,3,1,2,3,4) and not the ring finger (4) on E and the thumb on F# (1,2,3,4,1,2,3)
So, of the rules you are talking about I only use the one for the thumb and only for scales, but not necessarily for chords, triads and arpeggios. (how would you play a F#major arpeggio F#,A#,C#,F#->A#,C#,F#,A# -> C#,F#,A#,C# ->F#,A#,C#,F#???)
About the rule not using the pinky finger for black keys I have never heard, but it makes sense to play in front of the keys and so it makes sens to use the longer finger to play the black keys as they are further behind.
It also make sense to use the pinky finger that it doesn't get spoiled and lazy to stay in form and strong ;)
I've just seen in your example:
It seems to be a Boogie or Blues like pattern in the left hand:
For beginners it might be an advantage to use the same fingering for the F and Bb chords.