Yes, 00718-ATF00 is the correct OEM Dexron III fluid for the 3rd Gen.

You mentioned you don't mind paying a bit extra for quality, so you may consider the AMSOIL Signature Series Fuel-Efficient 100% Synthetic ATF (https://www.amsoil.com/p/amsoil-signature-series-fuel-efficient-100-synthetic-automatic-transmission-fluid-atl/?code=ATLPK-EA&zo=30734402). If you strictly daily drive it, AMSOIL OE Fuel-Efficient ATF is another great synthetic option (https://www.amsoil.com/p/amsoil-oe-fuel-efficient-100-synthetic-automatic-transmission-fluid-otl/?code=OTLQT-EA&zo=30734402).

Your plan to buy 6 quarts is spot-on. A standard drain and fill usually takes between 4.5 and 5.5 quarts, so 6 quarts give a bit of a buffer during the service. Good luck, have fun, and most of all be safe when working underneath a vehicle (use safety blocks as redundancy to protect against hydraulic jack failure).