Hi!
As the title says, I’m looking for a good book on English punctuation.
Preferably something that has an easily browsable dictionary/index and perhaps describes punctuation problems by types of text (letter, dialogue in a novel, dialogue in drama, titles, etc.).
Oh, if it notes differences between British and American usage, that would be perfect. 
The Chicago Manual of Style may be more than you need, but it’s an industry standard.
You can subscribe online or buy the hardcopy edition.
Thanks, I’ll definitely give the free trial a go.
I consider The Bedford Handbook by Diana Hacker to be the final word on American punctuation. For British, I go with Eats Shoots and Leaves by Lynne Truss.
Strunk & White’s The Elements of Style is my go-to book for this kind of thing. I’ve been using it since I was like 11 years old.