I'd like to customise the login form. Now, before you start typing "Make a file called Security_login.ss" as is written absolutely everywhere, I did that, and it's not what I need. I don't need the page AROUND the login form to change, I need to customise the login form ITSELF. So whatever is in `$Form` is what I need to modify.

Where can I add a template to do that, what should I name it in order for SS to pick it up, and what options do I have as far as templating goes?

I'm using SS 3.1.13.

Quite a hostile way you have of asking for help.
https://docs.silverstripe.org/en/3.1/developer_guides/security/authentication/
&& http://api.silverstripe.org/3.1/class-LoginForm.html

I do? I didn't mean to be hostile. Just trying to be as clear as possible, that's all. I'm trying to prevent replies like "make a Security_login.ss and put $Form in it", because that's literally all I got in my searches and it seemed like sort of defacto standard reply for questions like this.

Anyway, I don't know that I need to write a whole new login class. I'm not trying to change the functionality in any way. Just the HTML is what I need to have changed...

So... you don't want to change the template; you just want to change the template?
Make a template for MemberLoginForm(.ss) - see the default Form.ss template for details.

I appreciate you're trying to be clear, and it helps to include detail. But there are softer ways to approach the same end linguistically.