This seems to be making heavy weather of fairly straightforward issue, but here we go.
When a wave is formed, it is stable. It has a particular wavelength and a particular wave height in a combination that gives it stability. If the wavelength reduces, or the height increases, the wave gets taller and thinner until it can no longer stand up by itself and it - literally - breaks. The top falls off.
Why would the wavelength decrease? Because shallow water slows the front edge and the rear catches up. Or because tide, ocean current, or estuary outflow do the same thing.
Why would height increase? Because out-of-phase wavetrains run into each other and one wave 'climbs' on top of another.
There is nothing odd about calling it a breaker as the wave does in fact break, and most resources give around 1680 as the first recorded use.
The truth of the wave breaking comes from hydrodynamics. Anybody who has swum in the sea, or been on a boat, knows that the only motion in a stable wave is vertical. A swimmer just rises and falls - he/she does not move with the wave. Now look at surfers riding their breaking waves. For them, the water has movement and they do move with the wave. That movement is the top of the wave falling/breaking off.
The foam blown off a wave by the wind is a different thing altogether.