1. Please explain and simplify the 1st two paras herefrom?
The choice of prepositions depends upon the temporal context in which you're speaking. "On ~ afternoon" implies that the afternoon is a single point in time; thus, that temporal context would take the entire afternoon as one of several different afternoons, or in other words, one would use "on" when speaking within the context of an entire week.
"In ~ afternoon" suggests that the afternoon is a temporal space in-and-of-itself, wherein anything that happens will happen amongst many other events. In other words, the temporal context for this usage would be if one were speaking of a single day -- whether past, present, or future -- and of a single afternoon, during which many things might happen.
In your example,
the afternoonof that day = (on) that day's afternoon
= (on) that day, in the afternoon = in the afternoon (on) that day.
I don't know a rule for it. Sometimes only "on" fits the context, sometimes only "in", sometimes no preposition is allowed, and sometimes we have a choice...
The prepositions in the alleged sequence of equalities confuse me. 2. The first phrase uses
on the afternoon, but which contradicts
in the afternoon in the last phrase?
3. If you start with the first phrase and proceed forwards, then how did on capriciously change into in? If you start with the last phrase and proceed backwards, then how did in capriciously change into on?