The first method is how you actually calculate the forward price of a specific bond. You need to use the repo rate for that bond as the financing rate inside the calculation.
The second method is a quick way of estimating bond forward yields, but it is not something you can execute in practice. For example, if you try to lock in the yield , 5yrs from today, of a 10year zero coupon bond , the method assumes you will buy a 10year ZCB and sell a 5yr ZCB. But during the first 5years, the financing costs of being long the 10yr and short the 5yr do not perfectly offset. You have to borrow the 5yr bond in order to short it, for which you provide cash collateral receiving some interest rate. This may not match the cost of financing the purchase of the 10yr.
The second method is more appropriate for interest rate derivatives , whose implicit funding rates do offset perfectly. It is also commonly found in textbooks about bonds, but as I said it is an approximation.
Last comment : for a specific bond, there is a one to one correspondence between forward price and forward yield. It is just simple bond math. If you know one, you know the other.