:)

You should be able to connect a monitor to the laptop with the broken screen.
 
If that's not feasible, you can pull the drive and put it in an enclosure to make it an external hard drive or use a docking station like this one.
 
https://www.amazon.com/Thermaltake-BlacX-N0028USU-Docking-Station/dp/B0012Z3MKW

The short answer is No.  You can't just swap the drives.  One of the reasons is the OEM Windows OS.  It is only legally licensed for the PC it was delivered with.  Second, there are driver files on the disc for the specific hardware, which can be (and usually is) different on two different machines.  
 
What you can do, if you want to swap the drives, is first make a disc image (of the entire drive all partitions) and save that to a separate drive.  Then, restore the HP disc image to the 500GB drive and restore the Lenovo to the 1TB drive.  This requires another separate USB connected drive to do this.  You may not have another drive but you really should be making full disc backups anyway so getting one is not a big issue.  You can download the free version of Macrium Reflect disc image software to do it.  Do not use "Clone", its iffy to start with (some have no problems with it and others it never works), stick with the disc image.  
 
Macrium Reflect Free

togo
You should be able to connect a monitor to the laptop with the broken screen.
 
If that's not feasible, you can pull the drive and put it in an enclosure to make it an external hard drive or use a docking station like this one.
 
https://www.amazon.com/Thermaltake-BlacX-N0028USU-Docking-Station/dp/B0012Z3MKW

Don't swap the drives and try to boot another machine with it.  It probably won't work, and you could corrupt the drive.
 
The previous suggestions are good ones!
 
1. The external monitor trick is an easy solution, should work well, and give you plenty of time to back your Lenovo data up to an external USB drive.  Plus, since your system still boots up it's not yet time for more extreme measures. This would also probably be a good time to make a system image with Macrium for safe keeping of all your data.
 
If booting was an issue, I would proceed with the next step. 
 
2. Use a USB external enclosure or dock for the Lenovo drive.
 
I like this one, by Inatek.  It works with 2.5", 3.5", and 5.25" drives, plus it's SATA and IDE compatible.  This is a great tool to have for copying data to another drive or system.  https://www.amazon.com/Inateck-Converter-Adapter-Optical-UA2001/dp/B00MLYV5XY/ref=sr_1_13?s=pc&ie=UTF8&qid=1507218287&sr=1-13&keywords=inateck+usb+3.0

If you've got data on the drive (that you want to access), you can put it in a USB docking station... and connect it to any PC (to move/copy that data).