So the lower-ish difficulty answer would be to run the iso installer in a VM with the usb stick forwarded to that VM.

Or you can learn what those fancy installers do: on debian you would use debootstrap

Here seems the whole guide on how to install debian manually with it:
https://gist.github.com/tr3buchet/6407920

Btw, this is also basically how you install Arch. As of until recently there wasn't any installer and you had to go through each step manually (create partitions and fs, install the base system with <insert distro specific tool>, chroot, update fstab, distro specific finishing touches, voilà)

Copy a live cd iso using dd.

I used to have a SanDisk Extreme Portable running Ubuntu. If it was unplugged, my computer would boot Windows and when I plugged in the SSD to USB it would auto boot into Ubuntu. ~~I have no idea how I did it though.~~ It was my first time using Linux and I followed a guide online.

Edit: found the video

I installed from one flash drive with the image (on ventoy) to another flash drive that was plugged in to be the boot drive. On a cheap USB2 drive, it's unusably slow - so make sure you use the fastest drive you can

PenDriveLinux or rufus or balena etcher (frequently just referred to as "etcher") or just dd.

Rufus. Best for unlocked Windows images btw.

Ventoy.

without having to reboot to run the installer?

I'm not sure that I understand what you mean. Are you saying that you want to be able to load the OS without having to reboot your computer? Or are you saying that you just don't want to have to click the equivalent of "try the OS" when booting a live USB? If it's the latter, you should be able to just select the flash drive as the install point (though, tbc, I have never tried this, but I don't see why it wouldn't work) (I think you'd need 2 USBs, though — you'd need 1 to be the installer source, and one to be the install point — I don't think theres any installer that can run as a desktop application. Though, if it's Arch Linux, you might actually be able to call pacstrap from the host OS — I've never tried this after having already installed the OS). There's even OS's that are specifically designed to be ephemeral on hardware in this way — eg Tails OS.

Yes

MX Linux (debian based) works perfectly on a usb drive

Fatdog is Debian by way of Ubuntu designed to run from a USB drive. There's documentation on the website.

directly onto my USB

directly without having to reboot to run the installer?

You use "directly" three times. Remove all instances of the word from your post and reread it. Does the post make sense to you still? Does it have the same meaning?

I am not trying to be a dick, I want to make sure the word does not have a meaning I am not aware of in this context or if Linux is installable to a USB drive 'indirectly' but that does not make sense to me.

Can you rephrase what you are trying to do?

Buy an external ssd or an internal one with an enclosure.

MX Linux and it's predecessor (can't recall the original version) is a Debian distro that will run with a persistence cache on a USB stick.

Don't do that

You can but it will be very slow and your drive will die quickly. Alternatively you could make a USB drive with MX Linux and then only save what you need.

Linux mint will install and run from a usb drive as long as you unmount it upon loading its live version. Then it will allow to install on it during the installation procedure. I have an old Mac Mini and an old Macbook Air running Mint 22 that way.

Yes. Just partition the drive manually, install packages with debootstrap, bind-mount /proc, /sys and /dev, chroot into it and install a bootloader. If you don't understand what I say, you have to run an installer, possibly in a VM.