Terminator, afaik, doesn't have exactly what you describe, with the buttons... however it does have a right-click menu for custom commands. I don't really make use of this feature but it seems just what you are looking for, aside from the step of right-clicking to get there. The benefit to this method is that you can have (seemingly) unlimited commands in this list! It's also a powerful terminal with GUI to make most things easy, but also capable of some more advanced voodoo.
A simple yet powerful example use case: in my window manager (i3wm) I have a key combo that launches new desktop with quad terminal all grouped together, so I can, for instance, press alt+g and then ssh in to my server on all four of these terminals at once (and NOT on any other terminals I have open since the are not in the group). After I'm in I just alt+d and group input is disabled again. This is just one of like a bazillion features in Terminator. Aside from the specifics of the i3wm part of the equation, the the quad grouping is a breeze to configure in the GUI's layout editor.
Before I used i3wm, which is a tiling WM, the killer feature in Terminator, for me, was the ability to split the terminal horizontally and vertically via mouse or kb shortcut. If you don't use a tiling WM already that might be a game changer for you too.
Terminator isn't the most light-weight terminal (although it's certainly not bloated, it's just not as light weight as something like 'st') and on a tiling WM much of what I describe can be done with other terminal emulators - that said, I still use Terminator because I find it so much easier to get the layouts I want with it, and to easily launch layouts with different color themes, groups, etc. Making it much nicer when you are working on multiple servers to keep it obvious what's what.
Just my two cents. Hope you find what you're looking for!