The meanings can overlap.
To "quote" is to repeat someone else's words, in a way that indicates that you are repeating someone else's words, as opposed to incorporating them into your own work without any explicit identification that they are copied. (This could be plagiarism or it could be an allusion, but that's a different issue.)
To "cite" is to reference some other work. You may be quoting it word-for-word or you may simply be referring to it. "Cite" can be used to mean "quote", but it would rarely be used if you gave a quote without a reference, and it can be used when you give a reference without a quote.
Examples: "As Winston Churchill said, 'I have nothing to offer but blood, toil, tears, and sweat.'" This is unquestionably a quote. It might be called a citation.
"According to Barclay's Famous Quotations, Winston Churchill once said, 'I have nothing to offer but blood, toil, tears, and sweat." This is both a quote and a citation.
"In his History of Britain, Charles Stover says that Churchill called on the British people to be willing to make sacrifices." This is a citation but not a quote. It is not a quote because we are not repeating any exact words.
And to be complete: If I wrote a novel, and at some point in the novel I write, "General Framnitz urged his soldiers to fight on. 'I have nothing to offer but blood, toil, tears, and the billions of dollars of loot that we will get when we take the city'", that would be an allusion and not a quote, because I am not making any clear reference to Mr Churchill, and because I'm adapting the quote for my own purposes.
A citation can be informal like my examples here, "As he said in ...", or it could be a more formal reference, like "See Stover, Charles. History of Britain. Fwacbar Publishing, 1964, pp 85-86" (The exact formatting varying depending on what style guide you're using, if any.)
Best Answer
The prepositions by and with are both considered prepositions of agent.
That's to say, they can both refer to situations where someone/something acts on someone/something else.
Which preposition you prefer depends on the context.
Your example:
is correct.
To say: Don't distract yourselves by those phones sounds like beside those phones.
An internet search for prepositions of agent will offer you numerous examples of where either by or with is the correct choice.
Beware, however, of using prepositions of agent when you are talking about instruments.
One frequently comes across media examples of people saying things like: killed by a sword or shot by a gun, suggesting that swords and guns go around killing people.
There was a famous example when a BBC journalist asked a British politician after a political fracas whether he had been struck by a chair.
No, said the politician, correcting his questioner's English, he had been struck by a man with a chair.
http://english.edurite.com/english-grammar/preposition-of-agent.html