The use of the comma is a matter of style, and there are as many places to put a comma as there are styles.
The Chicago Manual of Style, for one, addresses your situation in the second scenario (two independent clauses) below:
6.32: Commas with a participial or adverbial phrase plus a conjunction
When a participial or adverbial phrase immediately follows a
coordinating conjunction, the use of commas depends on whether the
conjunction joins two independent sentences. If the conjunction is
simply a part of the predicate or joins a compound predicate, the
first comma follows the conjunction (see also 6.23).
We were extremely tired and, in light of our binge the night
before, anxious to go home.
The Packers trailed at
halftime but, buoyed by Rodgers’s arm, stormed back to win.
If the conjunction joins two independent clauses, however, the comma
precedes the conjunction (see also 6.22).
We were elated, but realizing that the day was almost over, we
decided to go to bed.
Strictly speaking, it would not be wrong to add a second comma after
but in the last example. Such usage, which would extend the logic of commas in pairs (see 6.17), may be preferred in certain cases for
emphasis or clarity. See also 6.26.
Source: The Chicago Manual of Style (login required)
The ball fell to the ground, and after it bounced, he caught it.
The ball fell to the ground, and, after it bounced, he caught it.
The ball fell to the ground and, after it bounced, exploded above the hot sidewalk.