have a chip on one's shoulder: be touchy or embittered (from a former US practice of so placing a chip as a challenge to others to knock it off).

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Excerpted from Oxford Talking Dictionary
Copyright © 1998 The Learning Company, Inc. All Rights Reserved.

This is the meaning given by the Chambers 21st. Century English dictionary; it doesn't square at all with Armorel's version, IMHO; it is far closer to your other answer, and that's where my points would go. Here is another "American chip":

Questions & Answers: Chip on one's shoulder. What is the origin of 'chip on the shoulder'? ... Jessica Ronaldson in the USA: “I was wondering what the origin of the phrase chip on the shoulder was?” ...
www.worldwidewords.org/qa/qa-chi1.htm

So it seems that our friend Russel(l?) was a bit of a wit, and had a little joke at his own expense.

In England this expression is always used either jokingly or sarcastically. It always means, that s/he is more than one way disatvantaged.
The "well balanced" is a figure of speech, as there may be two things against that person, both shoulders are equally laiden, therefore you could say "balanced". But it doesn't mean a well-balanced person, if anything, the opposite.
"...But who wants to have a chip on both shoulders?"
Definition of a well balanced Spurs fan – A guy with a chip on both shoulders…
"...we were working for Conservative and Labour Ministers – so we’ve got a chip on both shoulders."
"One of my wonderfully witty friends insists he's perfectly balanced as he has a chip on both shoulders!"
... Who put a chip on both your shoulders?
...I had a new boss who was a well-balanced individual-he had a chip on both shoulders and more hangups than your average art gallery...
"You can always tell an Englishman, by the chip on his shoulder. The Irish are much more balanced race, we carry a chip on both shoulders." (Dave Allen)
"Jake is a man with a chip on both shoulders – angry, confused and unanchored."

There seems to be a difference between US and UK usage - in the UK, as has been pointed out, the phrase doesn't commonly have its original sense of belligerence, wanting to pick a fight etc - rather, it's used of somebody who feels hard done by, moans about his lot , has a sense of inferiority etc. In the US it may be used more in its "looking for a fight" sense.

"to a have a chip on your shoulder INFORMAL

to seem angry all the time because you think you have been treated unfairly or feel you are not as good as other people:

- "He's got a chip on his shoulder about not having been to university".

There seems to be a difference between US and UK usage - in the UK, as has been pointed out, the phrase doesn't commonly have its original sense of belligerence, wanting to pick a fight etc - rather, it's used of somebody who feels hard done by, moans about his lot , has a sense of inferiority etc. In the US it may be used more in its "looking for a fight" sense.

"to a have a chip on your shoulder INFORMAL

to seem angry all the time because you think you have been treated unfairly or feel you are not as good as other people:

- "He's got a chip on his shoulder about not having been to university".