Which computer should I get for gaming purposes? Refurbished HP or new Lenovo? Links in update?

Get a alien computer they are the best

For gaming… Try Alienware but a good unit with an SSD drive would work. (go for an i7, if you are serious about gaming).
However, a serious gamer would not be using a laptop.

For gaming, laptops generally are not the way to go. You want something with a good GPU. Most laptops have low-end integrated graphics processors. If they do have a dedicated GPU, they can still perform worse than what you see in a good video card on a desktop PC.

This makes no sense what-so-ever for anyone that actually understands them.
Neither is capable in modern gaming.
The HP is an expensive overpriced old system, intended as compact in size
The Lenovo is a cheap system at good value.

Lenovo is A12-9720P, 1366 x 768, has 4GB Soldered ram and a 4GB ram memory stick apparently, but that is not clear.
Has 1TB HDD, no SSD

A12-9720P is a 4029 score cpu
It has a mid-grade R7 Bristol Ridge integrated graphics
That beats all Intel integrated that is not in the IRIS Pro categories, and is raked at 243

Some modern games can be played at low settings.

The HP shows an i5-6200U
That is a 4Q2015 CPU, almost 3 years old.
Benchmark at a similar 4014
Graphics is only Intel HD 520 ranked down at 393, and the laptop is 1920x1080 as needing a stronger GPU, so this is awful at gaming.

The Lenovo is far better than the HP at gaming, weaker at internet performance because it lacks an SSD for internet cache, but equal in CPU performance.

If you have under $500, the Lenovo is far better.
You might look into the process of changing an HDD to SSD by cloning the hard drive and swapping.
Gaming laptops have improved for price-performance on a sale.

$540 Acer Laptop Aspire 5 A515-51G-58SA Intel Core i5-7200U (2.50 GHz) 8 GB Memory 256 GB SSD NVIDIA GeForce MX150 2GB GDDR5 15.6" Windows 10 Home 64-Bit 15.6" Full HD 1920 x 1080 15.02" x 10.35" x 0.85" 4.85 lbs.
4641 CPU score, rank #126 graphics card, does have an SSD

https://www.newegg.com/...6834316369
at $560, it swaps out the i5-7200U for i5-8250U
This shows a 7649 score, and is a far stronger laptop and a great bargain for the price.

Next step up is $700 for a GTX 1050M graphics processing laptop.

If you can't get into the $560 number, the Lenovo is a bargain priced laptop for what you get, but don't expect miracles out of an under $400 laptop

The HP is not to be considered in gaming at all.

Neither will work for any sort of gaming. They are both garbage.

They are both not good for gaming. The first one has a good processor but onboard graphics. It is more of a business laptop then gaming laptop. But it will probably run some modern games on very low settings. But i wouldn't recommend it. The second one is a cheap low-end laptop. The processor is very outdated. It uses 6+ year old architecture. The amd graphics might be able to play some games decently on low setting but i would not recommend its not very future proof.

if you want a good fairly future proof gaming laptop get something with a gtx 1060 and a core i5 or core i7 processor.

https://www.newegg.com/...gnorebbr=1

I never buy HP computers--total garbage whether laptop or desktop, no matter what era (since 1992, and never got better).
Lenovo is a cut rate Chinese company that a decade ago purchased all IBM Thinkpad business and proceeded to destroy that reputation.

Neither are going to be great, but the Lenovo would be better because it's got an AMD A12 processor which has a much better graphics core than the Intel processors.

Gaming, stay away from refurbished. If you truly want to game, it's worth the extra money.

The phrase "Business Series" don't bode well, Lenovo's out of stock, can't help.

The HP is way off, unless your idea of gaming is just Portal Flash and Portal 1, in which case anything today can cut it with room to spare; anytime you see Intel HD without specifying that there's a hybrid Quadra/FireGL/GeForce/Radeon then stop looking, it won't play any blockbuster games. Which is true for the to that HP so it is out. Lenovo is a tad better by Radeon R7 but I do not know how good that is in an absolute sense.

I have an old Core i7 that does have hybrid Quadra NVS in it; didn't buy it for that since the eBay seller I got it from did not list it just the primary Intel HD. But I'm glad it does, though I'm not quite sure how Windows knew to switch to Quadra when I start a game.