Totally agree that (successful) startups go through a three-stage lifecycle. Many others agree with the “three stages” approach as well. Steve Blank famously calls them “Search/Build/Grow” but I find Jeff Bussgang’s three-stage metaphor the richest and easiest to understand and communicate.
(Described here: http://www.inflexion-point.com/Blog/bid/30202/The-Jungle-the-Winding-Road-and-the-Highway — I think he’s referring to Jeff’s initial metaphor. and I wrote my take on Jeff’s metaphor here to explain why I like it so much.)
I think it’s still important for startups to identify themselves by and refers to “Seed”, “pre-seed”, “Series A/B/C” etc. for other reasons. It’s important for investors to tailor expectations about funding round sizes and valuations, and vice versa. It’s a way to signal what type of funding the company is likely seeking, and what amounts and valuations an investor is typically willing to entertain or provide. And also is a way to roughly benchmark companies of similar rounds against each other- “you guys are looking for a $4m Series A at $20, but I just saw another company doing a $4m A at $20 with a more exciting market and twice the revenue” etc. So I still think the round-identification terms have value, just — as you say — not necessarily to identify the lifecycle stage of the startup, but to identify the funding stage of the startup.