When I press the phone key, my contact list appears, touch the search box and as stated typing 2-3 letters gets me my target.

But just pressing instead the search icon in the desktop will bring up the keyboard and then type the 2-3 letters.. My preference...use the keyboard while in desktop, and do not even need to press the search icon. Just start typing and your contacts appear. Emphasis here is that the Search tool is very nifty.