I got the second test file above, 
http://literaryfashion.com/ssi-test/index.php , working. The missing piece was turning on short_open_tag in domain > PHP Settings.
But HTML files are still not being parsed by PHP, no matter what combination of PHP version / PHP handler / Apache directives I try.
For PHP version I have mostly settled on "5.4.16 by OS vendor" -- other options tend to cause .php files to be downloaded instead of parsed.
For PHP handler I have tried
  FastCGI application served by Apache
  FPM application served by Apache
  mod_php served by Apache
(The latter after first enabling support in Tools & Settings > Apache Settings "5.4.16 by OS vendor mod_php"-- but it does not seem to work. The help link "view the php_info() page" at domain > PHP Settings results in a blank page.)
With FastCGI, I have tried the Apache directives given above:
<IfModule mod_fcgid.c>
   <Files ~ (\.html)>
      SetHandler fcgid-script
      FCGIWrapper /var/www/cgi-bin/cgi_wrapper/cgi_wrapper .html
      Options +ExecCGI
      Require all granted
   </Files>
</IfModule>
No parsing of PHP in HTML files.
With FPM application, I have tried essentially all combinations of 1, 2, and all 3 of the following Apache directives:
  AddType application/x-httpd-php .html
  AddHandler x-httpd-php .html
  AddHandler php-script .html
Nothing results in PHP parsing HTML files.
Now what?