My site used to be http without SSL. I recently installed SSL to https. I went into the admin settings and changed the setting from http to https. When my site opens in Firefox on a PC and a Mac it opens as http. When I open it on a Mac in Safari, it opens as https.

What can I do to set some consistency here?

I believe FireFox and Safari indeed behave different.
What you now need is a "force redirect" to https for every request.
Please ask your host how to proceed because every host has its own way of doing this.

Need to edit the .htaccess file and do something like:

<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /

RewriteCond %{HTTPS} off
RewriteRule ^(.*)$ https://%{HTTP_HOST}%{REQUEST_URI} [L,R=301,NE]
(rest of file)

But, if you specifically start with https in the address bar it should keep using it. The first three lines should already be in the file. Just add the bottom two below them.