As you point out there are several Titebond wood glue products. Any of them (that I am aware of) should work fine for a PDP. Elmers Carpenter's Glue is what I use. Elmer's Glue All is fine. Gorilla Wood Glue (the white stuff) is good. Gorilla Urethane glue is not.
Titebond II is the one for record cleaning. OK this is off topic but since I mentioned it. I hear Elmer's works but is more difficult to use. The idea is that the vinyl records and the ??poly-vinyl-acetate?? (not sure of the formulation exactly , this is from long lost memory) are very similar and won't interfere with each other (ie.won't melt your record). And
please try this on some old dirty thrift store record first, before you do it (which you may eventually) to some cherished piece of vinyl. Don't come back here blaming me for ruining your record collection, cause you did them all at once and used the wrong glue and forgot and let the glue sit 6 months.
Here's the deal. You want to spread a thin even layer of glue over all the grooves. I've seen old broken record players used as free spinning platters (lazy susan) to aid in spiraling a bead of glue from inside to out, but this was for doing a lot of disks. So in the end you have a spiral of glue, and you can do it by hand on a stationary record. Then the glue is spread to coat the grooves. Use a business card or credit card etc... you end up with a film of glue. Don't squeegee it all of, leave a film above the peaks of the disk.
That explains the glue, but before you start applying the glue get some of the blue masking tape. Cut 8 one inch strips. Apply them to the blank edge of the record (the run in) and to the inner blank edge (the run out). These are pull tabs, I'll explain in a minute. When you spread the glue be sure the glue goes over these tabs.
Set the finished record aside on a level surface (not the dash of your car). At this point heat and humidity and air circulation will determine how long you need to let the glue 'dry'. 4-8 hours. There is a point when the glue will be stiff enough to remove as a solid single piece sheet. If you let the glue set too long (days) it will become brittle and crack and be more difficult to remove as it will be in small pieces. Elmer's is known to be more brittle (but work if you get the timing right). Titebond II is the favorite and has more flex and longer time frame to remove it.
OK so you found a copy of Tiny Tim at the Salvation Army. And it is totally gross. Somebody spilled coca-cola on it in 1967 and it's never been cleaned. And you are dying to hear him. First things first use dishsoap and luke warm water and clean hands. Soak and lightly scrub with your fingers to remove the worst of the grunge, rinse well. Do this over and over till you just can't get it cleaner. Use your standard record cleaner now. The fluids and brush will probably find more stuff in the grooves that you missed at the sink. Now that you have it as clean as possible (normal in home cleaning) you can try it on your record player.
Still nasty sound, even with this much cleaning? You can take it out for professional cleaning. $$$
Or you can give this old school cleaning a try.