What if asterisks were used for <em> and <strong>, and underscores for <i> and <b>. This would preserve backwards compatibility visually, while allowing authors to make the appropriate distinction when necessary.
E.g.:
I really like The Big Lebowski!
I *really* like _The Big Lebowski_!
(Note that this may introduce accessibility issues, where things that were properly emphasized in legacy documents suddenly are not properly emphasized. However, I’m guessing legacy documents already had accessibility issues with strings that were emphasized inappropriately.)