In case I send John Smith an email and he sends me a response, should I include "Hi John," when I am replying to his email?
I always do. This is a matter of personal preference. Some people view e-mails as electronic letters, whereas others view them as text messages. I belong to the former group, so I always include a greeting like "Hi John," or "Hello John," or even "Dear John," if it's a formal e-mail. I also always include something like "Best regards," or "Sincerely," at the end of the message before signing my name. I'm slightly turned off by e-mails that start without a greeting. Others don't care one way or the other.
A lot of business e-mails these days will just begin with the name of the person being addressed; e.g., "John:". I personally find this a bit brusque, but it's widespread.
Bottom line: whether you include a greeting is a matter of personal preference, and it's perfectly acceptable to use the same greeting in subsequent e-mails.
When in Rome, do as the Romans do. Meaning, watch how others are using or omitting greetings in email, and tune your style to theirs.
Also, take into account the tone and context. Is this a friendly exchange? Academic? Business? In business, emails among colleagues and equals, who don't have anything to prove to each other (such as "I am not a threat to you"), one can often be more terse without causing offense.
In practice, what often happens is that when there is some quick back and forth on a particular topic, the greeting often gets dropped after approximately two messages from each party, but the one-word signature often remains; but this also gets dropped after another couple of rounds of back-and-forth.