If I take money out of my RRSP and put it back in an RRSP with a different bank, is there a penalty? To avoid the penalty, I guess there needs to be a straight RRSP --> RRSP transfer?
If you withdraw money from an RRSP, you are taxed on that money as ordinary income and will have to pay withholding taxes, and potentially more income tax when you file. Additionally, you lose that contribution amount.
If you subsequently deposit the same gross amount into another RRSP at a different bank, that will lower your taxable income for the year and you'll subsequently be able to recover the withholding taxes, but you will have permanently lost the contribution room.
For example, you have $100,000 in RRSP contribution room. You buy $1,000 in RRSPs from Bank A. This leaves you with $99,000 in RRSP contribution room.
Next year, you withdraw that $1,000 and deposit it in Bank B in an RRSP account. Now, you have $98,000 in RRSP contribution room. Chances are good that Bank A withheld 15% for taxes, but when you file your tax return, you'll get that back. Your withdrawal and subsequent purchase of RRSPs in the tax year cancel out.
Far better is to transfer the RRSP from Bank A to Bank B. That has no tax implications and would not affect your RRSP contribution limit. Not all banks are willing to do this, though.