I've been advised by my new host's tech support that my options are (their words):
(1) Change the nameservers on the existing domain, park it, and route it to the site on the new server, or
(2) Have the primary domain name changed (I've no idea what this might mean - the domain name is the domain name - my words), or
(3) Transfer the domain to the new host.
I'm not sure what the implications are between these options. I feel inclined to ask my current hosting provider (the one I'm leaving and the one who originally registered the domain) to arrange (with the registrar they use) to replace their nameservers with the nameservers of my new host, and to keep the domain with them (and pay their renewal fees when they come up) rather than put all the eggs in the one new basket, at least for the time being.
In terms of search engine rankings, especially Google, and maintaining my traffic, these days does it matter which way I approach it? And is there a "best way" to handle the transfer? I've searched here on WW and read that the site should be left on the previous server for a while (not a problem), and even to have the domain pointing to two nameservers (didn't know this was possible or even sensible, given that I'm moving hosts because of bandwidth charges).