What you have written is right, but it is just the first step.
The release and solvation of an electron is fast. Reaction of an electron with ammonia to release hydrogen $$\ce{2 e-(solv) + 2 NH3 -> H2 + 2 NH2-(solv)}$$ is slow. Deep blue (diluted) or bronze (concentrated) "sodium electride" solution in liquid ammonia slowly converts itself to the colorless ammonia solution of $\ce{NaNH2}$.
For the water case, all is extremely fast, as water is a much stronger acid than liquid ammonia. Blue-like color can be catched by 10000 fps camera, what is commented in SE Chem Q - Reaction of potassium with water
BBC article, referring the Nature article by Pavel Jungwirth and colleagues (ref.1),
ABSTRACT: Alkali metals can react explosively with water and it is textbook knowledge that this vigorous behaviour results from heat release, steam formation and ignition of the hydrogen gas that is produced. Here we suggest that the initial process enabling the alkali metal explosion in water is, however, of a completely different nature. High-speed camera imaging of liquid drops of a sodium/potassium alloy in water reveals submillisecond formation of metal spikes that protrude from the surface of the drop. Molecular dynamics simulations demonstrate that on immersion in water there is an almost immediate release of electrons from the metal surface. The system thus quickly reaches the Rayleigh instability limit, which leads to a ‘coulomb explosion’ of the alkali metal drop. Consequently, a new metal surface in contact with water is formed, which explains why the reaction does not become self-quenched by its products, but can rather lead to explosive behaviour.
with high speed (cca 10000 fps) camera photos shows the blue colour near 0.3 ms after the metal/water contact (ref. 2). It may be rather more toward violet, it seems to me. The abstract of ref. 2 explains the blue color:
Alkali metals in water are always at the brink of explosion. Herein, we show that this vigorous reaction can be kept in a non‐exploding regime, revealing a fascinating richness of hitherto unexplored chemical processes. A combination of high‐speed camera imaging and visible/near‐infrared/infrared spectroscopy allowed us to catch and characterize the system at each stage of the reaction. After gently placing a drop of a sodium/potassium alloy on water under an inert atmosphere, the production of solvated electrons became so strong that their characteristic blue color could be observed with the naked eye. The exoergic reaction leading to the formation of hydrogen and hydroxide eventually heated the alkali metal drop such that it became glowing red, and part of the metal evaporated. As a result of the reaction, a perfectly transparent drop consisting of molten hydroxide was temporarily stabilized on water through the Leidenfrost effect, bursting spectacularly after it had cooled sufficiently.
References:
- Philip E. Mason, Frank Uhlig, Václav Vaněk, Tillmann Buttersack, Sigurd Bauerecker, Pavel Jungwirth, “Coulomb explosion during the early stages of the reaction of alkali metals with water,” Nature Chemistry 2015, 7, 250–254 (https://doi.org/10.1038/nchem.2161).
- Philip E. Mason, Tillmann Buttersack, Sigurd Bauerecker, Pavel Jungwirth, “A Non‐Exploding Alkali Metal Drop on Water: From Blue Solvated Electrons to Bursting Molten Hydroxide,” Angew. Chem., Internat. Edn. Engl. 2016, 55(42), 13019-13022 (https://doi.org/10.1002/anie.201605986).
