As ghettocottage pointed out (and I have put in these forums myself more than once), the formula to search from *almost any* {not just google} search engine is going to be -
site:url <space> search term
I'm sure you see your mistake at the moment Craig, and why you got the results you did, by doing the search like it was an address, but I'm just as sure that someone will come along a while from now and do the same thing you did, so hopefully this helps out those people.
Now I will include this handy graphic -

- In days of olde....
..., followed by some search engines that are not Google but that do respect your privacy such as
http://duckduckgo.com/ - includes "!bangs", which let you search using other engines if you really want to. I don't use those much.
https://lite.qwant.com - this is the 'lite' version of Quwant.com that I use, the full version is a little heavy for my taste but you really should decide for yourself.
https://www.startpage.com - There used to be two sites like this, the other was lxquick, however far as I know, this is the one remaining version. Billed as "Google results without the privacy issues".
As to the OP, I'm not sure what to say there. I don't have Mojave to test with, but with 10.6.8 Snow Leopard and Safari, I certainly see the search without being logged in.

- A little old and slow, but still functioning....
I've never accessed the forums from this machine previously, and so have never signed in from it. I would suspect that means it has no cookies or other method of knowing who I iz, so I can only assume this is how it would be seen universally.