Short run means any activity or plan of less than one year. And long run is any activity or plan longer than one year. Its the difference of time. Strategies of firm depends on their short run or long run planning. The tactics of profit maximization for short run would be different than long run. For instance, In short run, a firm might bear a loss in order to cover fixed cots. But In the long run your losses should be marginalized by your profits. If they are not you are running inefficiently.
And yes,,, Profit=total revenue-total cost
And yes,,, Profit=total revenue-total cost