This is a very interest discussion. I think it is due to the difference between Mortgage and APR (annual percentage rate).
1. The most accurate compounding is 'continuous compounding' which used in option valuation, such as Black-Schole Model. The formula is: e^rate.
2. For daily compunding as the mortgage providers clami, it should be
(1+daily rate)^365
3. For monthly compuning will be exactly as you said above, (1+monthly rate)^12
4. However for mortgage payment calculating, it is the MORTGAGE RATE used, not the APR:
PV annuity = Amount x (1/r) x [1 - 1/(1+r)^n]. The annual rate / 12 is used.
That is why mortgage providers give both mortgage rate and APR.
Thus, Excel's PMT and IPMT are correct.