I am using 1.9.3, so I'm assuming that's a new enough version (?).
I'm convinced it is something related to either the config.php file, or something else not causing the upgrade files to be doing their thing. You simply cannot take a 1.9 config.php file and dump it into a fresh 2.0+ moodle root dir and expect the upgrade process to kick in, and I'm baffled as to why this isn't mentioned in the upgrade wiki. The 2.0+ config.php file has extra function calls in it, for a start.
However, if I use the config-dist.php code provided in the 2.2.7, replacing the values as necessary to mirror those in my current config.php, the upgrade doesn't start, it just starts a standard installation. I've tried deleting various /install/ and install.php files before starting this process, but it also makes no difference.
I see you mention "Moodle needs to see the old version database and moodledata directory" - does this mean I should alter the new config.php file to reference the moodledata directory that is sitting in the OLD dir.? Again, the wiki does not mention you have to do that, but is it worth a try...?
eg:
$CFG->dirroot = realpath('/home/DOMAIN/public_html/2.2.7-moodle');
$CFG->dataroot = realpath('/home/DOMAIN/public_html/1.9-moodle/moodledata');
??