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The one I linked should be fine. Here it is:
It only takes a second to put it up or down, no screws required. Pull on your door trim with your hands to make sure it's secure before putting the bar up.
Most any will do. I have this one and it's fine.
What you need to worry about is the door frame itself, not necessarily the bar.
Make sure your frame is the correct width for the bar.
Make sure there is enough clearance around the door frame. Some doorways won't work if it's right next to a wall.
Make sure the frame of the door is solid. Pull on the frame with your hands and fingers and see if the frame wiggles or is loose. If it's loose you could damage the frame when you attempt a pullup.
Mind giving me an example? The kind I had was like this one. It put too much weight on the upper frame of the door so even just pulling on it a bit would cause it to creak, so I didn't trust it.
I had this one in WV and it was fine
Ig it depends on what kind of pull up bar you want, if you want one of those that go on the door frame (e.g. https://www.amazon.com/ProSource-Multi-Grip-Chin-Up-Pull-Up-Black/dp/B01H7OW5KY/ref=sr_1_6?crid=1K8L0OI9NGITA&dchild=1&keywords=door+frame+pull-up+bar&qid=1597268900&sprefix=door+frame%2Caps%2C210&sr=8-6)
then you'd probably need to keep the door open whenever you use it, which is really just your call (which is not a big deal if your in apartment style, bigger deal if your only doors open to a shared bathroom and the hallway).
A pull-up bar isn't really something you want to cheap out on, but here are a few good choices.
The third one I sent you is much nicer but it's out of your price range. If you can find one with that design for cheaper, I highly recommend it.
These types of bars are more sensitive to doorframe style, so make sure to measure all your frames carefully.
> Can anyone recommend a basic pullup bar that I could use in my house
Depends on your house, if you have a regular and sturdy door with some space around it, this kind is cheap and will work just fine. With whatever you decide to get, check the dimensions and watch some youtube videos, like this one, to make sure it will fit.
> but might very well give up by next Tuesday?
Nonsense, don't give up. Just get something and ease into it to make good habits.