If you read any contemporary published plays (Annie Baker, Steven Levenson, Stephen Karam, to name a few), you'll know that they'll use a slash (/) to indicate when the next line should begin. Who was the playwright that pioneered this notation?
David Mamet was very much into this, but used ellipses and dashes like most playwrights. His “follower” Neil LaBute did it a lot as well and may very well have popularized the used of forward slashes.
It’s less one playwright’s influence though, and more the gradual use of fully naturalistic dialogue, which slowly throughout the 20th century replaced the more “poetic” style of writing that was once more commonplace.
If it's one character breaking in I use a simple dash. If a few words should/might be overlapping, that's for when it reaches rehearsal. I don't fine tune it to that degree. If it's two characters talking at the same time just do double columns. But the only rule that counts is to make it as clear as possible and anyway you see fit to do so is fine. You're the playwright!