Case 1 is where the Stock Price is GREATER than the Strike Price, not the other way around as you have stated. In this example a net gain will result if St < K as well, which demonstrates that an arbitrage opportunity exists when Ct − Pt > St − K*e^−r(T−t).

I think this is where your logic goes wrong:

"...To that money that we owe, we add the money that we owe to the contract buyer.."

K is not referring to the money that we owe the contract buyer, but rather the money we receive from the contract buyer at the exercise price. Yes, the call is in a losing position, however K is just referring to the money received at the strike price.

So, the equation is really saying that the money we borrowed to finance this strategy (plus the accrued interest) is less than amount we ended up receiving when the contract buyer exercised his option and bought the stock at the strike price. We end up with a net gain.