Thank you. Good idea!
I ran traceroute and discovered that example.com resolved to the IP of an old server I once used.
Checking my registrar, I found that I had left example.com and www.example.com resolving to that old server's IP.
Deleting those entries has resolved the issue. When I check example.com it shows the expected "You are seeing this page because the DNS record of example.com is set to this server's IP but Cloudron has no app configured for this domain."
Thank you for your help!