If I use the telephone to transfer money from my statement savings account to my checking account, does this count towards my six allowable transfers? My bank says it does, but I thought it didn't count since I was transferring money into my own account at the same bank.

Your banker is informing you correctly. Telephone transfers do count toward your limit. The rule to remember is that if it doesn't require you to get up and go to the bank or an ATM, the transfer will generally count against the monthly cap.