How is filing a contact under "Phone (Google)" different from the others? I thought managing contacts in Android would be the easiest of tasks, but I have found myself in doubt now that I have an Android phone running Android OS 2.3.6 (Gingerbread).

If I file a contact under "Phone", I can search it and group it, but it is not synced to Google.

If I file a contact under "Google", its synced but it is not groupable (from the phone) and sometimes searches skip it (depends on whether it is in Google+ Circles or filed under "My Contacts").

How is filing a contact under "Phone (Google)" different from the others?
Adding a Contact under Google lets you sync it with your Google account. I'm also able to add a group to a Contact, which is also synced with the Google account. 

However, I'm no longer using the stock Contact app. Instead, I'm using TouchPal Contacts, which allows me to hide contacts without a phone number, something the stock app doesn't have.