A bit of background (feel free to skip):
I've installed Linux Mint on the SSD of this computer before as dual boot setup that would ask me whether I wanted to boot into Mint or Windows 10 Pro when I powered on the computer. However, when I later removed Linux Mint, the computer still kept booting to GRUB on startup, and the BIOS kept recognizing multiple boot sources simply called "ubuntu," with absolutely no way to remove them. I resorted to wiping the disk entirely and getting a clean Windows install.

My problem now:
Well, I want to start using Linux Mint on this computer again, but I'm scared the same thing will happen. This time, I'd like to install Linux Mint Cinnamon v20 on the HDD (rather than the SSD containing Windows) of the computer, and I don't want a dual boot. I want the computer to boot to the SSD with Windows by default, and only go into Linux Mint if I enter the BIOS and select the HDD with Mint as the boot source myself.

About my computer:
HP EliteDesk 800 G1 Small Form Factor
Intel i7 4770
32GB RAM
GIGABYTE GeForce GTX 1650 Low Profile 4G
1 TB SSD (contains Windows 10 Pro)
1 TB HDD (empty)

So, my question is...
If unplug my SSD, boot to a live USB with Linux Mint Cinnamon v20 to install to the HDD, then plug the SSD back in after the installation, will booting into the installed Linux Mint (from the HDD) modify the SSD at all while I'm using the operating system? (except for when I go into Mint's file explorer and access the SSD myself)

I've done a fair bit of searching and haven't found anything similar on the forums, but my Google-fu admittedly isn't the best. I apologize if this is a bad/repeated question. That being said, if anyone can answer my question, I want to give you a big thank you in advance!

You will need to choose from your BIOS whether to boot from the SSD or the HD to select the appropriate OS.
You should set your boot priority in your BIOS so that your most frequently used OS comes before the other OS.
For example:-
1. USB
2. Mint [HD]
3. Windows [SSD]

When you switch on you will get a couple of seconds to press some key combination to allow you to override the default choice.

Mint will not modify the SSD at all - unless you choose to tell it to mount a SSD partition and write to it.
There are some advantages in actually doing this.

Presuming a [U]EFI system with Win10, your original problem is simply solved by changing the default boot in the NVRAM on the motherboard. Nothing to do with the SSD or hard-disk. So your concern is largely misplaced. But the Mint install will become the default, and will over-ride the Windows SSD - regardless of whether your take it out or not.
Best solution is to leave it in, and let grub build a dual-boot menu. If you take Mint or the hard disk out in the future, simply reset the default in the firmware to the SSD.

Let's say you have HDD-1 and HDD-2. If Windows is on HDD-1, you can unplug that harddrive, leave HDD-2 plugged in, and install Mint on that (HDD-2). Mint will install the grub bootloader to the MBR of HDD-2. After that is done, plug HDD-1 back in, set the bios boot order to boot from HDD-1 first; and you are good to go. At that point, u can bring up the BIOS boot options, and select HDD-2 when you want to boot mint. If you do that, yes, as you guessed, the windows boot loader won't be affected/overwritten by grub, as it would have been initially. You don't even have to unplug HDD-1 actually, if you know what hard drive to install the boot-loader on; but, unplugging it is a safe gauruntee you won't accidentally overwrite the windows boot MBR (Master Boot Record). GL.

Nope. Win10 requires [U]EFI and gpt - the old MBR over-write issues are consigned to history.

+1 for syg00

When you say that Mint will “over-ride” the Windows SSD, do you mean that it’ll simply go to the top of the boot order list? If so, that can easily be changed from the BIOS, of course. I apologize if I’m getting to caught up in terminology. I just want to make sure I 100% understand what I’m doing before going ahead.

Also, having GRUB make a dual-boot menu is not optimal for me as this is unfortunately a shared computer. I need the computer to boot straight to Windows on startup so that other users don’t have to go through any extra steps. All of of my Linux stuff will ideally be relegated to the HDD and only accessible if I deliberately boot to it from the BIOS.

Hi, thanks for the detailed reply! I might just be getting caught up in terminology here, but when you say Mint will “over-ride” Windows, do you mean it’ll simply go to the top of the boot order? If so, that can easily be changed from the BIOS, of course.

Also, a dual-boot menu is not optimal for me as this is unfortunately. a shared computer. I need all my Linux stuff to be relegated to the HDD and only accessible if I deliberately boot to it. That way, other users will go straight to Windows without extra steps.

Yes it is simply an order manipulation. It can be done from the firmware, or using efibootmgr (from Intel) as a terminal command in Linux.
If it were me I'd simply install Mint then change the default back to the SSD - when you want Mint hit F10 or whatever your motherboard uses to get the hardware boot list up to choose from.
Simple.

And to save getting yourself and others confused, I'd advise you to stop using the term BIOS - I like firmware as it covers both BIOS and EFI and (hopefully) makes people stop and think before interjecting.

Alright, I installed Linux Mint on the HDD and Windows still works as normal on the SSD. I can’t thank you enough for the help!