Seconding, here, that I've almost-always heard arbitrary/arbitrarily to mean (slightly differently than the sibling commenter) "able to be chosen freely, without any loss of generality"; though, yes, I have also heard your usage of "by personal judgement, disregarding established rules or convention".
In fact, I just had cause to use the term again, which prompted me to return here to describe my usage a little bit more.
I was documenting a configuration file, and noted that one particular value was chosen "arbitrarily". What I meant by that (and what I am reasonably confident that my audience would understand) is that my choice of that value was not influenced by any other concerns - I did not have to make sure it matched some other value in the file (or in another configuration value), or that it conformed to some enumeration of permitted or significant values.
So - yes, I was technically stating that the value was "the result of a judgement or choice", but _much more strongly_ I was stressing the fact that (personal) judgement was the _only_ factor in the selection, not conformance to other prior values, rules, or restrictions. So - not purely random, but in some sense "closer" to random than to pre-determined.
I agree with you here (https://news.ycombinator.com/item?id=37731820) that entirely throwing away the meaning of the word "arbitrary" and making it a synonym for "random" would not be useful - but I don't think that's what's proposed. Arbitrary is (and, I believe, always has been) a word which stresses the importance of choice and judgement - but sometimes that means "strongly considered judgement was applied here to select the optimal option", and other times it can mean "any selection which you choose is probably satisfactory here - there are no hidden constraints, rules, or requirements".
