I like the Iron Gym Pull Up Bar on Amazon, works great for me and seems pretty durable so far. Just make sure it will fit your doorway and wont damage the door frame!
Does anyone have any suggestions/advice for choosing a good pull-up bar?
Get the door frame-mounted kind, not the single extending bar kind. The former is much safer and can handle more weight.
Note that you should strictly be using it for controlled vertical movement, it's not designed for horizontal movement.
What do I look for?
Confirm the core materials used, you want metal over plastic for the bar frame.
It's arguably okay if the handle padding is shit, you can remove it and buy grip tape at an athletics store.
Make sure you have a buffer zone between whatever max load you'll be inflicting and the rated weight it can handle.
Double and triple check the measurements of the pull-up bar with the available space in the door jamb.
And will it damage my door frame?
Not unless it was constructed improperly and/or has prior damage. Most door frames are designed to withstand way more weight than you could add. That's why they're safety zones in earthquakes and stuff.
It might mark up the frame lip moulding (easily scrubbed off) or dent it if the moulding is weak, so you can cut up some old, thin socks or towels and slip those on the pull-up bar over the points of contact. Careful not to use thick fabric that will displace it.
I've seen ones at Wal-Mart and thought about getting one of those but its Wal-Mart and I don't know how good they are.
You don't have to shell out too much for a decent pull-up bar, maybe $30 or so.
This is a popular choice: https://www.amazon.com/Iron-Gym-Total-Upper-Workout/dp/B001EJMS6K
I'm 6'6", in the UK our doorframe standard height is exactly that, 6'6" (199cm). I have a doorway mounted expansion bar. The major drawback is the week or so of head trauma that results from walking into the thing after first installing it, until the muscle memory duck manoeuvre becomes habitual. Technically, the drawback of an expansion bar is that I cannot really train power because I'd faceplant into the doorframe/wall above the bar. But, I can close the door without having to take it down & put it back up etc.
Doorframe mounted - https://www.amazon.com/Iron-Gym-Total-Upper-Workout/dp/B001EJMS6K/ref=sr_1_5?crid=KB18V8A80XEK&dchild=1&keywords=pull+up+bar&qid=1604949837&sprefix=pull%2Caps%2C273&sr=8-5
This type gives you the clearance of the doorframe/wall to do power but, you cannot close the door with it in place. It also puts all of the weight/momentum downward on the doorframe (not what they're designed to withstand) - this is the most likely of these solutions to damage/break your doorframe.
Chin-ups are fine, pull-ups are fine. With my hands gripping the bar to begin either of those, I am at rest on my knees at max scapula extension. This is fine for just those exercises. I do them with my ankles crossed behind me, floor to top and back.
Anything involving leg movement where your legs are at some point hanging freely at rest is completely out. That bar you linked goes to a max height of 88.18" - just over 7'4". For your height, you would almost certainly require a bar that is at least 8' high to be able to hang and also let your legs hang relaxed, without touching the floor.
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TL;DR - If all you plan to be doing is pullups/chinups then a doorway mounted jobby is fine. A tower is better in that you can also do other great exercises on it (eg dips), so if you have the space for one get one. If you want to involve your core (free hanging legs) then you'll need a tower with a bar that is higher than 8' and at that point it is your ceiling's height that you need to worry about.
I have this one (or something like it) where the bolts stick out on the back side, I cut up a towel and taped it to the hardware so it doesn't crush the drywall.
Get a pullup bar for one of your doorframes so you have something to do pullups on and then girth hitch these to the bar when you want to use them.