The final verdict is in!
Thanks to everyone, I have moved the site to the new domain (actually it's duplicated but that's OK since I can erase the old one). When I started to act on the recommendations, I sort of accidentally discovered a 4th option (explained below) which worked. But I wish I'd done it the way suggested by @Robin S, @Autofahrn and @jens.martsch (updating $config->httpHosts in site/config.php). I'm sure that would have been even faster.
So anyone else who comes across this thread looking to do the same sort of thing (moving site to new domain on same host), you might like to try that solution of $config->httpHosts in site/config .
What I wound up doing
However, I had not yet read those 2 latest responses at the time, so I was thinking about all the different methods and was looking to try the Duplicator module. Desiring to be a responsible site owner, I went to create a Softaculous backup of my site before trying the Duplicator module. This way I can restore my original site in 1 click if I do anything wrong. To my happy surprise, in the backup menu in Softaculous, one of the other options besides back up was "Clone site". Well, being curious, I thought "Let's try that, and if it doesn't work out, I can use one of the other methods people suggested". So, I tried Clone Site, and it worked perfectly! I just had to do point and click and tell it which of my domains to put the cloned copy of the site in, and boom, there it is at the new domain! After doing that, I then read the newest responses to my thread and realized that it would have been even faster to use the solution of Robin S, Autofarn and jens.martsch, but anyhow, the Softaculous method works too in case anyone has it available to them and was wondering about it.
And after the site is in the new domain...
A big thank-you to pwired for the advice given
This turned out to be really helpful - I tried the search function in phpMyAdmin and found a few mentions of my old domain name still in there. It was clear from the results which pages I needed to edit in the processwire editor. So it may be a useful step for others too who are looking to do a similar type of domain move.
Thank you again to everyone who responded in this thread. All of the suggestions were very useful. I really appreciate the sense of community here.