Agreeing with that description of “bitter”.
I think of salty as something like “sarcastically nasty” – but possibly with good reason, and possibly being funny about it. ‘That old lady got really salty when he told her not to worry her pretty little head about it’.
Exactly this. They are not synonyms to me, though they’re certainly closely related. More a pseudo-synonym. This difference is subtle, but I think of bitter as being more extreme and longer-term. Salty is more annoyed, bitter is more festering anger. “He’s a little salty because I just beat him at that game” vs. “he’s a little bitter because he always gets beaten at that game.”
“By accident” is still almost thirty times as common in printed English as “on accident”:
I wouldn’t have expected otherwise.
That’s the definition I’m familiar with in California. I would never use it to describe someone who is resentful or angry. You can call some people salty and they will even take it as a compliment.
There’s supposedly a similar generational divide between “Believe me…” (older) and “Trust me…” (younger).
Both feel common and natural to me. I can’t say offhand if I use one more than another, but they have slightly different meanings to me.
Same here.
As for by accident/on accident, “on accident” makes me switch (born before 1975).
This and I first heard it in California in the late 80s.
I’m down with that. 
Neither “believe me” or “trust me” was that popular before about 1979 or 1980 compared to how popular they are today. “Believe me” is still more popular. It’s not that really that much more popular:
If you restrict them both to sentence-starting instances (via a case-sensitive search), “Trust me” takes over around 2011-2012:
When I hear “trust me” the first thing I think of is “Trust me. I know what I’m doing” from the 80s series Sledgehammer. It should also be noted that what is popular in edited, wrtten copy may lag or just be different than how a language is spoken in its environs.
“Hear me now and believe me later.” Late ‘80s Hans and Franz made “Believe me” sound kind of ridiculous.
SALTY DOG By Papa Charlie Jackson, Paramount 12236, Chicago, c. August 1924.
Lord, it ain’t but the one thing grieve my mind,
All these women and none is mine.CHORUS: Won’t you let me be your salty dog,
I don’t wanta be your man at all,
Salty dog, you salty dog.
I don’t know if salty has the same connotation here, but 1924, anyway.
From etymonline:
Someone who’s bitter about something might go off to sulk quietly about it.
Someone who’s salty about something won’t.
I first saw “salty” meaning annoyed or angry when I first started lurking on Twitter about a decade ago. It was very popular there, especially (it seemed) among the younger crowd.
For me “salty” has the connotation of “acknowledges that it’s a bit silly to be upset about this but is annoyed anyway,” like, “I’m still salty that the waiter took my dish when I was saving the best bite for last,” while “bitter” has more of a connotation of “this was upsetting and it’s still upsetting to the person,” like, “He’s still bitter over the divorce.”
The exception is the phrase “Not that I’m bitter,” which functions very much like “salty.”
As my father would tell me, “Don’t say ‘Trust me’ - it makes you sound like Richard Nixon.” I’m not sure where he got that from, but it stuck.