all I can come up with is that the Modem (router ? ) that we use at home feeds both computers, one via a hardwire Ethernet cable and one WiFi, and it is the IP number of that gadget that the outside World is noting, not individual computer numbers
Correct. The public IP address of the router, assigned by your ISP, is what the website is looking at - plus cookies, potentially.
should we take a computer to California then our IP number recorded would be the one relevant to Starbucks WiFi outlet, if that is what we might use. N'est ce pas ?
Again correct. You could achieve the same effect by asking a neighbour or friend if you could drop in and use their internet connection - or by using a 3g / 4g device instead of your ISP internet connection.
You may be able to change your router's IP address by switching off your router for several hours until the DHCP lease expires (assuming it's dynamically assigned by your ISP, and the lease period is in hours, not days - outside your control, however).
I'm confused because I've played around with ExpatShield and the like,which assigns an IP number to my computer - or does it do it to the Modem - that bears no relation to my present location
No, it does neither. It simply proxies your connection, relaying your traffic to the target website via another router / computer in the remote country location. That would work once, but again the IP address would become known to the booking site. It's possible that your ExpatShield IP address might change, but only within a relatively small pool.
You could look into anonymiser providers, such as TOR, but you are getting into slightly murky waters. They would give your internet connection a seemingly remote egress point - much like ExpatShield - but randomly at some TOR exit point somewhere in the world, and this exit point would probably be different each time you used it.
SD