It's the (awkward) shift+middle-click combo that brings up Terminator context menu for me when putty_paste_style = True.

This is in contrast to the docs, both the terminator_config manpage and the online manual, which state middle-click alone will bring up the Context Menu.

$ terminator --version
terminator 1.91

$ cat /etc/lsb-release 
DISTRIB_ID=Ubuntu
DISTRIB_RELEASE=18.04
DISTRIB_CODENAME=bionic
DISTRIB_DESCRIPTION="Ubuntu 18.04.1 LTS"

The mentioned shift+F10 combo also works, but also stuffs garbage (;2~) onto the command line.

The settings in gsettings set org.gnome.desktop.default-applications.terminal exec doesn't affect the behavior of the context menu entry. The used terminal is gnome-terminal and that's hard coded in libterminal-nautilus.so in the package gnome-terminal. And this library is used by Nautilus.

Perhaps this will be changed in future versions, but currently it is a fact.

1. Install Nautilus-actions in 18.04
   Launchpad user Daniel Marynicz has created PPA for Ubuntu 18.04 LTS with Nautilus. Follow those:
   sudo add-apt-repository ppa:daniel-marynicz/filemanager-actions
sudo apt-get install filemanager-actions-nautilus-extension
   but you need use fma-config-tool to launch.
2. Add action to nautilus

after those steps, in the right-clicked menu it will show 'open in terminator' like this

but it is still inconvenient！
3. Set action to the top level in menu
open the config of the nautilus

then, deselect "create root ..."

4. enjoy it