All of this is in the vim on-line help. But it's distributed across the doco for a number of variables and options, and there's not really an overarching guide to the whole thing.
Normal mode
Your terminal emulator has two modes. In its normal mode, it itself handles all mouse interaction. When you double-click, the terminal emulator decides what a word is, and does the highlighting of your selection. When you middle click, the terminal emulator obtains the selection from the X server and sends it to the terminal as if it were ordinary typed input.
If you run up minimal vim (which is the program with the name vi), or use :set mouse= in full vim, this is what you get. If you've ever tried pasting into vim using the middle mouse button you'll realize that vim is clearly not involved in the paste, because if you aren't in insert mode at the time the pasted text is treated as vim commands.
Mouse reporting mode
In its mouse reporting mode, however, your terminal emulator does not handle mouse actions itself. Rather it transmits the actions as if they ordinary typed input, encoded as special control sequences. An application program outputs other control sequences to turn mouse reporting mode on and off. (There are quite a lot of subtleties to this that relate to the fact that some of the mouse reporting protocols invented by the free software world were really bad. But that's beyond the scope of this answer.)
When you have mouse=a or mouse=h or similar set in vim, it changes the terminal mode on the fly between normal and mouse reporting modes, depending from the exact setting of mouse and what you are doing at the time. For example: When mouse=h, it turns mouse reporting on when you switch to a help screen, and back off when you switch away again.
When mouse reporting mode is on, it is vim itself that is deciding what a word is, that does the highlighting of your selection, and that communicates the selection to and from the X server. If you run up (full) vim with mouse=a or similar but without a way to know that there is a GUI available, by removing the DISPLAY environment variable from its environment, you'll find that whilst selection and paste work within vim, it cannot see the X selection and X applications cannot its selection. Indeed, there can be two selections on the screen concurrently, the one in an X application and the one in vim. (Minimal vim does not have the ability to talk to an X server in its code at all.)
Deciding what is a word
The rules for deciding what a word is differ. Your terminal emulator, depending from what emulator program it is, may or may not let you configure how it determines word boundaries. roxterm, Terminate, and LXTerminal all have a "select-by-word characters" setting in their preferences dialogues. Others simply hard-wire this.
Whichever is the case, it clearly thinks that a/b/c/d is all one word. So when the terminal emulator is in charge of dealing with mouse actions, this is what it sets as the selection in response to a double-click.
vim has its own set of quite independent settings, that are based upon what is considered to be a "keyword" and thus link into the regular expression matching and syntax colouring as well. Clearly whatever you have vim configured to understand about the current file (the iskeyword option being local to a buffer and adjustable by your selection of filetype) considers a/b/c/d to be multiple words separated by slashes. So this is what you get when vim has assumed charge of dealing with mouse actions.
Further reading
:set mouse?in Vim?