I was under the impression that starships use a combination of fusion and M/AM.
True. Fusion is primarily for STL & AUX Power, M/A-M is for FTL & Primary Power
Ds9 had 3 large fusion reactors for example, but not M/AM reactors... I would imagine that if SF was able to upgrade those so that DS9 was able to power a plethora of powerful phasers and shields which can fend off over 50 ships, it stands to reason fusion has been improved quite a bit and that M/AM is mainly used for Warp (but as we saw, its also used to power replicators onboard and other systems when necessary).
But DS9's Fusion Reactors are several size orders larger than the typical Fusion Reactors we see on StarShips.
The main issue with Fusion Reactors is the need to scale the containment vessel to produce the energy output necessary.
On a Space Station that doesn't move, you can easily scale up the reactor size as needed and generally be ok.
If you're moving FTL, than that Fusion Reactor size affects the housing of your vessel and it's support structure, which affects the size of the Warp Bubble necessary & affects the mass that you have to move at STL.
M/A-M reactors are rather slim when you look at it's profile and is linear, so it's easy to figure out how to place it in a vessel.
So, Earth never really transitioned from one to the other... they kept using both and improving them at the same time.
From my understanding, they developed both technology lines simultaneously, and realized one was better to use for StarShips, the other was better to use for stationary platforms.
The Phoenix was a different thing though... it was meant as a way to test out Warp drive (relatively short flight duration)... so while its likely it never used Dilithium, it is probable that it used an advanced form of fusion which was advanced into an early prototype for a Warp core. Dilithium came later on as a way to stabilize/regulate antimatter reactions.
I highly doubt Zeffram Cochrane had access to Dilithium or they even knew how to make a working M/A-M reactor at the time.
It was probably Fusion power that got him to his first Warp flight.