When I hear green dial I think of Seiko Alpinist (SPB121).
Phhhhuh
The obvious answer is Hamilton Khaki Field "Murph" 38 mm, or Sinn 556 as you're already looking at.
With regards to Sinn, I think 556A (with or without red second hand) has more personality than the 556I, the numbers make a nice break from the otherwise extremely sterile dial, and it also makes a stronger tool watch look. You should note that 556I has a glossy black dial and 556A has a matte dial, so they're not identical. The downside is that 556A doesn't dress up equally well, but to be completely honest no one but you will ever care about that these days — people will see a black dial on a leather strap and think it looks fancy enough. Hamilton Murph dresses up a lot better of course.
Either Antea or Marine by Stowa, there are a few different options. Antea is a bit dressier. Their white dials are actually silver, they look white in most lights but can look silver or creme as well, and looks great with the temperature-blued hands which also take on different shades in different lights.
No logo, no date for me. It looks cleaner, and the brand is stated somewhere after all, like on the back of the case.