The terms long pre-date public railways, going back to the 1600s. The first usage of "railway/railroad" was, IIRC (can't see my copy of Lewis from here..), in the first half of the 1600s (pre civil war, I think) with "waggonway" being used from the 1680s at least. In the northern coalfield there were also "wainways", which were dedicated cart-roads for haulage of coal. Some of these were converted to (railed) waggonways, some supplanted.
There doesn't seem to have been an early preference for "way" vs. "road" - of the two first public railways, one (the Lake Lock) was a Railroad, the other (the Surrey Iron) a Railway. It's possible the proximity of the SIR to London helped the -way form to become more popular.
In south Wales in the 1780s-1810s "tramway" and "tramroad" seemed to be used interchangably for plateways, too - there doens;'t seem to have been a preference based on length or public access. In fact, one of the longest of them - and one which operated a passenger service - was the Brecon Forest Tramroad.
Of course, these //are// Americanisms ;)
Actually, it's the process of technology transfer of railways/roads to the americas is quite a disputed area, with different researchers giving different emphasis to expats, military engineers (the first railway/road in the americas probably being built by the British Army in colonial times), american visitors to England (e.g. Strickland) and transfer via published documents. The pioneering lines and the first steam railways in the US do seem to have been predominantly local efforts, albeit with expat, military, visitor-to-europe, published-document and artifact-transfer (locomotives!) playing important (though variable) roles. There's a good article on this in "Early Railways" (published by the Newcomen Soc.).