Which is anode and which is cathode?
Regardless of the polarity, the electrode where oxidation takes place is the called the anode and therefore reduction must take place at the cathode. The electron flow from the anode to cathode is as shown in your top picture. 

By way of an example, in an electrochemical cell, suppose that two beakers are connected by a salt bridge. In one beaker is a strip of zinc metal immersed in a $\ce{Zn(NO3)2}$ solution and in the other a strip of silver in $\ce{AgNO3}$ solution. The two metals are then connected by a wire and a current will flow. (The salt bridge supplies a return path so that the solutions remain electrically neutral). The redox of the Zn electrode is $-0.763$ V and that of the Ag $+0.799$ V. This means that electrons will flow from the zinc to the silver electrode. The zinc is oxidised to $\ce{Zn^{2+}}$ and the electrons are released into the metal and flow to the silver electrode through the wire. The zinc electrode is the anode and the silver the cathode.