MacOS – OS X updater claims installation is successful, but new OS is not installed

My guess from the details provided is that you don't need to apply any patches since Apple has patched bash as part of 10.9.5. If you really aren't on 10.9.5 yet, then keep reading...

Here is what an Apple-patched as of October 8, 2014 Mac should show from terminal:

mac:~ mike$ sw_vers
ProductName:    Mac OS X
ProductVersion: 10.9.5
BuildVersion:   13F34
mac:~ mike$ which bash &&  `which bash` --version
/bin/bash
GNU bash, version 3.2.53(1)-release (x86_64-apple-darwin13)
Copyright (C) 2007 Free Software Foundation, Inc.

(If you are new to terminal, just type the parts after ** mac:~ mike$** so:

sw_vers
which bash &&  `which bash` --version

After you've compared your values, you can quit the terminal app.

If your versions are not as above, I would delete your update files and start over. Download a new patch from http://support.apple.com/kb/DL1760 and then open the console app. Look on the left for a toggle button "show log list / hide log list" and show the list. Open the triangle under /var/log and click on install.log - set a marker and then run the installer package.

Watch that log for any errors relating to the manual patch.

As soon as a new Mac OS is released, the previous one becomes unavailable.

The only way to get an older OS directly from Apple is to have previously 'purchased it' - even if it was free - then it will appear in your purchase history in the App Store.

If it's not in your purchase history, then you'll have to get it from a trustworthy friend [or less-trustworthy torrent etc]

10.10.anything will do, as you can still get the combo updater to 10.10.5

It's always wise to get a new OS download, even if you don't intend to install it right now - just so it's always in your history. I have every OS back to Lion in my purchase history - good for if I ever need a disposable OS for testing etc.

As Sierra will not run on older machines, this rule has been relaxed for machines whose last supported OS is El Capitan.
You can download it from here - https://itunes.apple.com/app/os-x-el-capitan/id1147835434?mt=12 - but it will not run if your machine is capable of running Sierra.
If it is capable, then you are back to needing it in your purchase history already.