I'd be guided by the imperative of making the best sense possible for my reader. I'd hope that any rules that do exist were dedicated to the same outcome (with basic English grammar, they usually are; style guides might be a different matter).
With reference to the OP's examples, I'd suggest (with Michael Harvey) that you capitalise if you're referring to the singular, formally constituted "British Empire" (or any other particular empire).
Former proprietors of the Daily Telegraph may have enjoyed some of the ill-gotten gains of Empire.
If the slave trade was enabled by Empire, then the Guardian has begun to confront its own entanglements with that prefiguring imperial project.
But note: 1) and 2) only work if it's already understood that the delimiting context for either statement is "Britain".
If you intend a general meaning, don't capitalise.
3) Their Empire is, after all, just one example of empire that we could consider.
Note: 3) is just for illustration. Depending on where the sentence was placed, it might come across as wilfully contrary to leave off the "British" (or other nominal qualifier) like that.
Strictly speaking, use of the capitalised "Empire" on its own only tells the reader you are referring to one particular empire, and for your statement to make sense, the preceding text must already have made clear which one it is. If the stage is not set, sense-wise, then the capitalised "Empire" brings the reader up short, as they try to figure out which the heck Empire you're on about...
And that would be a Holy Roman f*** *p.