Origin of electrode potential and its connection with electric double layer
Let's see if we can tease out the main points here.  The images below are taken from Lance and Cole's Analytical Chemistry POGIL workbook.
Microscopic processes at a metal surface exposed to an aqueous solution containing the corresponding metal ion(1) will result in a slight negative charge on the metal surface due to oxidation of the metal and a slight positive charge in the region of the solution close to the metal surface owing to the sudden generation (and dissolution) of positively charged metal ions.

The amount of negative charge generated is metal specific and as of now, unknowable in an absolute sense.  Therefore, we are required to compare the extent of this process at one metal electrode to that of another.  If we compare the Zn electrode above to a Pb electrode, we would find that the Pb electrode has less negative charge on its surface relative to Zn.  Both of the electrodes, however, have a slightly negative charge at their surfaces.  Because the charge is greater at the Zn electrode, we say Zn is easier to oxidize than Pb.
To answer your question which of the above explanations is correct, the answer is both (although the clarity of the explanations is up for debate).  A negative charge is generated at both electrodes, but the the amount relative to one another differs, and that gives rise to the notions of easier/harder to oxidize/reduce.
(1) In my experience, microscopic theory of electrochemical processes typically assume the presence of the metal and metal ion in part to avoid issues which identifying what in the system is being oxidized and reduced.