As far as I know (I am both old and a native English speaker) "bawl" is one of those words most people know but don't use very much.
While its meanings are pretty much in line with your definitions, those definitions are much too general.
The most common use is to weep loudly and uncontrollably, as in the phrase, "bawling his eyes out", mostly with respect to children but also to an adult in extreme distress.
It is also used in some fiction to describe a character shouting while upset or frightened.
Back in the 50s, the Kingston Trio had a calypso song about a guy who met the grandmother of the girl he was seeing.
"Well the first time she hit me, she knocked me down,
Wouldn't even let me get off the ground.
I try to tell her of the rules of Queensbury,
She says 'Boy, that cuts no light with me.'
"And I bawled (I bawled),
I bawled (I'm sorry).
Oh man you should have heard me bawl.
Oh, never let me near a rolling pin
And that old lady again."
To "bawl someone out" is to reprimand them at length and probably at volume.
Cattle "bawl" when distressed.