By looking at paintings of 'old' naval battles I noticed that there is a change in where the ensign is flown that happens right before the Napoleonic Wars. If you look at this painting, you see that the ships fly their ensigns from the flagstaff but if you look at this painting, you see that the ensigns are now flying from the spanker. Why did this change occur?
You're seeing an inconsistency with how painters depicted the practice moreso than a change in protocol or anything like that. Ships of the period I study could and did fly ensigns from a lot of places, and sometimes flew multiple ensigns to prevent it looking like the ship had struck if one were shot away.
Now, that said, the flagstaffs that your first picture indicate existing certainly do exist, as you can see from this image of HMS Victory (warning, very large). But the flagstaff would often interfere with the working of the gaff boom, so it would usually be struck when the ship as leaving port, and the ensign flown from another spot -- sometimes the mainmast, sometimes the gaff, etc. A painter seeing the ship in harbor might assume she always carried her ensign on the flagstaff, and paint it that way.
Incidentally: In that picture, Victory is wearing the ensign at her stern, the flag of a vice-admiral (Nelson's last rank) at the mainmast and the jack on the jack-staff at the bow.
