Update from Date 6.2.2 to Date 6.2.3 unsuccessful

My attempt to install update from Date 6.2.2 to Date 6.2.3 failed. What I mean by that, my /admin/build/modules page show 6.2.2 version when I am absolutely positive that I have copied 6.2.3 code. Same with entry in the systems table. I have followed instructions (deleted date folder on the server before copying the new version and run update.php). When I run update.php, it does not list date module under "select versions". I have tried completely uninstalling Date module, deleting Date folder from the server, deleting entries in the system table (I didn't delete related tables), but the module comes back every time I access /admin/build/modules page (I can see the page if the Date modules are not installed)! And every time it shows 6.2.2 version. Frankly, I tried it few times, so I might have missed something in one of the attempts. This was not my first Date update - I have successfully upgraded to 6.2.2 in the past.

Question: how can I delete module? How can I upgrade to 6.2.3?

Some background that may relate to this issue: I have been suffering from Fatal error: Out of memory... when accessing /admin/build/modules page for quite some time. My provider 1and1 has limit of 40M. Since this error only happens on admin pages, I was able to work around it (installing more modules) by altering the system table and it worked. I have installed probably 5-6 modules this way, and my site was operating with no problem for months. I thought I found a magic fix. The same error happens when I run update.php, but most of the time database is either updated with tables or I just create them manually (I have separate test environment that I keep bare bones, so when installing new modules, I first try it there and get table descriptions from there).

I am using PHP Version 5.2.10.
Thanks!