this is interesting. in my experience it is always the third situation. they try to migrate the data slowly from "old system" to "nexgen". it never works. it sounds like maybe the answer is do a big bang switch over to the new system or don't rewrite at all. (i.e. the system you are rewriting probably needs to be small enough)
Ship of Theseus is the other way. Fact is a lot of companies have code that has been rewritten and they don’t know it, because the devs that did it know they would have been stopped if people knew. But piece by piece, bit by bit, everything was shifted to a new organizational paradigm from the original.
You can’t really do objective case studies on development processes that involve subterfuge. Even if they work.
Same company did a refactor for one app with great results. Waterfall style.
Same company did a rewrite that crashed and burned. Shipped the new app side by side the existing one as they gradually added features.
TL;DR Anecdotes