In English, if you want to say that you made somebody's acquaintance (1b), you say "I met them." You don't say "I became/got acquainted with them." So sagt man das halt. "I became/got acquainted with somebody" is used to mean that you actually got to know them (1a).
Examples:
Scenario 1:
At a party, A introduces B to C. B and C exchange a few pleasantries and a bit of small talk for a few minutes, and then part ways and mingle with others at the party and don't engage with each other any further.
B can say:
I met C at the party.
B would not say:
I got/became acquainted with C at the party.
Scenario 2:
B and C happen to see each other at another party a week later. This time, they sit down together and speak for a whole hour and they get to know a lot of things about each other.
B can say:
I got/became acquainted with C at the party.
although in everyday speech it's much more common to say
I got to know C at the party.
I went for a walk, and on the sidewalk/pavement near my house I met a strange woman... (begegnete ich)
Nein, das geht nicht, at least in US English. "I met a strange woman" would mean that you actually made her acquaintance. If you just saw her there or bumped into her or whatever, you would not say "met" in US English.
When I first became/got acquainted with him (or: made his acquaintance), he was still a very young man (als ich ihn kennenlernte)
No, see above. If you are just talking about the very first time you made his acquaintance (whether or not you actually went on to get to know each other), you would not say "got/became acquainted with him." You would say "met."
It's like there's two steps:
1.
make X's acquaintance /
meet X
2.
become/get acquainted with X /
get to know X
In each case, the red version is a more formal version of the green version, but the meaning is the same. The expressions in 1 do not mean the same thing as the expressions in 2. 1 corresponds to 1b in the Duden, and 2 corresponds to 1a in the Duden. You seem to think that the red one under 2 means or can mean the same thing as 1; that's not the case.
As for the examples from the WR dictionary, my guess is that they are using "become acquainted (with)" in a very literal sense as a way to indicate what meaning they are talking about. In actual speech, we would not say "become acquainted with" in this meaning.
I've never met your cousin. ≠ I've never become acquainted with your cousin.
Yes, we've already met. ≠ Yes, we've already become acquainted.
It seems that, when I wrote that ''meet'' is ambiguous, I wasn't wrong after all. / 'meet' ist doch mehrdeutig.
In what sense do you feel it's ambiguous (unless you've changed your mind)? Can you give an example sentence in which you think the meaning would be unclear (and there would be at least two possible readings)?
In any case, "meet" is an absolutely correct and idiomatic translation of "kennenlernen" in the sense given in the first post.