There is more to know about Tightbond Original, II, and III besides longer set times and the increased waterproofness with a higher number.
Titebond Original is an aliphatic resin
Titebond II is a cross-linking PVA
Titebond III is proprietary polymer
Titebond Original dries the hardest and will not clog sandpaper too much
Titebond II is a little more flexible
Titebond III has a very flexible bond, relatively speaking, the dried glue bead is never totally hard and will seriously clog sandpaper.
Titebond III also has the propensity to produce a raised glue line over time. Sand smooth, finish and 6 months later after a change in seasons you can feel a bump at the glue line. Sand or scrape down the bump, re-finish, and the bump will be back in another 6 months. As the wood moves with humidity changes, so does the glue. A good thing if you want a waterproof joint that will not open up. A bad thing if you want wood joints to stay super smooth. I use Titebond Original for all interior furniture, III is good for cutting boards and places that get wet.
Since I made custom entry doors for a living I needed the longest set time possible along with complete waterproofness. I used to use a resorcinol glue, it combined long assembly times with indestructibility. Gluelam beams made with this glue are still solid 70 years later. Unfortunately resorcinol was discontinued years ago, probably because a main ingredient was formaldehyde.
Polyurethane glue is a good substitute but not as strong, it also degrades in sunlight and extreme exposure. I ran years/decades long exposure tests using project cut-offs. The scraps were hung on a fence behind my shop to see what would happen. The resorcinal glue never failed, the polyurethane eventually did but it took a while. That said, the doors I made using polyurethane glue will still last 500 years.
Epoxy is just as strong as resorcinal, has gap filling properties, set time can be adjusted, can be used below 70°F, a lot messier and needs Acetone for cleanup.
Epoxy also degrades in sunlight and a high quality marine paint is needed if exposed to the elements 24/7. An ultraviolet absorbing varnish or coating can be used for occasional exposure to sunlight, like on wood canoes.
