Maybe the issue is connected with the site url you used when installing your site not being exactly the same as the url you now use to visit the login page (as opposed when you go via the register page as you are redirected to the site url).

With "not exactly the same" I mean that it could be a difference in "http" vs. "https" and/or in one case use "www" and not in the other way. This could explain why there's a mismatch in the session cookie created when login if the url in the cookie is not the same as the site url.

You can check the site url in elgg-config/settings.php. Then you can try if you still have the problem when using exactly this url in the browser. To get it working with different urls being used getting redirected to the site url you would have to add some rewrite rules to Elgg's .htaccess file, e.g.

# If your users receive the message "Sorry, logging in from a different domain is not permitted"
# you must make sure your login form is served from the same hostname as your site pages.
# See http://learn.elgg.org/en/stable/appendix/faqs/general.html#login-token-mismatch for more info.
#
# If you must add RewriteRules to change hostname, add them directly below (above all the others)

# http to https
# exclude subdomains
RewriteCond %{HTTP_HOST} ^(www\.)?SITE\.URL$ [NC]
RewriteCond %{HTTPS} off
RewriteRule ^ https://www.SITE.URL%{REQUEST_URI} [NC,L,R]
# add www on ssl
RewriteCond %{HTTPS} on
RewriteCond %{HTTP_HOST} !^www\.
RewriteRule ^ https://www.%{HTTP_HOST}%{REQUEST_URI} [NC,L,R]

That's what I use on my site, so you might need to adjust it as explained in the link included in the comment depending if you use www or not in your site url and use SSL or not on your site. In any case you would need to provide your site url in the rules.

Never used WAMP or XAMPP so I can only refer to info found via Google.

Does https://stackoverflow.com/questions/25922193/wamp-server-online-but-inaccessible-from-outside help?

Or https://www.google.com/search?q=WAMP+public+access with more links?

I saw this exchange earlier in the day but thanks for the offer.  What Apache system are you running?

Apache on Linux. I had used Dyndns for testing in the past and it worked fine. I got rid of Windows completely last year with a new computer (not really using Windows anymore on the old machine for years already and it was just annoying that OS updates from MS failed to be correctly installed due to the Linux bootloader). I never tried using the Windows installation as server as it seemed so much easier to do all that stuff on Linux.

If you still haven't got it working maybe someone else can help who has at least tried a WAMP or XAMPP installation.

In these cases, first you need to do is check your server error logs.

Also, don't ask about your problems if you runned Elgg on the localhost.

It's a very different situation instead of hosting.

"localhost" always refers to the computer you are currently using. If you used "localhost" as site url of your Elgg installation and try to access the site from the same computer the site is installed on it should work. But if you try to access the site from another computer "localhost" refers to this other computer so it does not fully work as the routing can't get resolved correctly.

If you only want to set up a test site on your local computer but still want to access it from other computers you could either set up a nameserver in your local network or make use of the hosts file of your operating sytem to resolve a "domain name" to the computer's IP addresses of your network. Works best (maybe only works) with fixed IP addresses used in the network. The domain name should be some fantasy domain, e.g. "mysite.home.nil" not used on the Internet to avoid issues. And you would have to enter the domain name and corresponding IP address on any computer in your network you want to access your test site from. Then install your Elgg site with the domain name you entered in the hosts file as site url. How to find your hosts file? Try Google (I have actually no idea because I've totally given up Windows).

If you want to access your site from a computer not within your local network it won't work with such a hosts file entry. In this case you would have to make use of some dyndns service. But as such a service is likely not to be found for free anymore you might think about hosting your site somewhere for real on a webhoster.