How can I get iTerm 2 to copy only what I select?

If you do a double-click to select it go into "all-word" mode, which automatically grabs each word.

If you just click and drag it'll take whatever your selection is.

If you go to Preferences ( ⌘ + , )

Under the Appearance Tab, there is an option "Show tabs in fullscreen by holding ⌘ for:" and there is a slider from 0 - 10 sec.

I had the option set at 0 sec and was getting the issue described here. When I set the slider to 10 sec, this was not an issue.

I like being able to see the tabs, but do not like it messing with my copy selection, so I went with 1 second, enough time for me to copy the text without it fudging with my selection and showing the tabs.

It is an option under Preferences>General>Selection. I am using iTerm Build 3.1.6.