Why are there two formats for the Solow capital evolution equations?
The Solow-Swan has the two versions, one is in discrete time (first equation) and other in continuous time. Both give the same results, but you cannot mix them - as you did. 

In the first equation (discrete time) you can take the first diference but never a derivative, since the variable is not continuous. The point above the variable means that it has been diferenciated with relation to time.

So the last equation does not make any sense because tried to mix both models.