OK, this seems to have worked. It may have been overkill, but nothing else seemed to do the trick:
1st site:
- placed admin.php and path.php in the root folder of my 1st site. In path.php, I set:
$system_path = "./myeesysdir/";
$cp_url = 'http://www.my1stsite.com/admin.php';
$site_name = '1stsite_shortname';
2nd site:
- in System Preferences > General Configuration, I set the CP url to
http://www.my2ndsite.com/admin.php
- placed admin.php and path.php in the root folder of my second site. In path.php I set:
$system_path = "../myeesysdir/";
$cp_url = 'http://www.my2ndsite.com/admin.php';
$site_name = '2ndsite_shortname';
Then I cleared all EE caches for both sites, and deleted cookies & cache in all my browsers (this seems to be crucial).
The CP login now works correctly for both sites.
Gayle