A facelift, a mini facelift, a neck lift and other facial cosmetic or plastic surgery terms are all very hard to sort out as to what they are exactly and what would be best for you.  The best facelift for you is the lift that deals with your issues and is directed at achieving your cosmetic goals.  From your photos, you have loss of facial volume resulting in a hollowness under your eyes (the so called tired look of aging), moderately deep nasolabial folds or smile lines as well as marionette, jowls and loose skin of you neck.  A simple lower facelift will address all of the issues.  You could always chose to have a less aggressive procedure but the downtime and complications are about the same either way and usually the longevity of the lesser procedures is not very good.  Some filler for the lower eyelid area would be of benefit and a good addition to any plan you decide upon.
Here is an article aimed at clearing some of the confusion surrounding facelift nomenclature.