I know that you can import a Font in Java with something like this:

File file = new File(fontPath);
Font font = Font.createFont(Font.TRUETYPE_FONT, file);
// alternative:
// Font font = Font.createFont(Font.TRUETYPE_FONT, new FileInputStream(file));
GraphicsEnvironment ge = GraphicsEnvironment.getLocalGraphicsEnvironment();
ge.registerFont(font);

Then you would use for example font.deriveFont(Font.PLAIN, 20); to get the desired style and size.

Example

But now let's look as an example at the font Consolas, there you have four TTF files:

consola.ttf (Plain)
consolab.ttf (Bold)
consolai.ttf (Italic)
consolaz.ttf (Bold & Italic)

Of course you could just import consola.ttf with the method stated above, but then using font.deriveFont(Font.BOLD, 20); isn't the same as using consolab.ttf because the plain font was simply transformed to look like a bold font.

Example Pictures

Here I used the installed font with new Font("Consolas", Font.PLAIN, 20); and new Font("Consolas", Font.BOLD, 20); (as a side note, if the font is installed on the system you also get the right bold font if you use deriveFont(Font.BOLD);):

And this is consola.ttf, imported with createFont and derived bold font (both with size 20 like the example above):

Well when installed it isn't a problem, but I don't expect others to have a custom Font, so I want to put the TTFs into the jar file, so that I can import them during the initialization via getResourceAsStream(path).

Is there a way to import all relevant TTFs and then just call new Font("Custom Font Name", fontStyle, fontSize); so that it's used like an installed font (Picture 1), and that it doesn't looks like a derived 'fake' bold font (Picture 2)?

I'm not sure what exactly is the problem. You got all your TTF files and you have to import and register them. Following tests use DejaVu Sans fonts which are not installed on my system.

Test 1

Font f = Font.createFont(Font.TRUETYPE_FONT, new File("dvs.ttf"));

GraphicsEnvironment ge = GraphicsEnvironment.getLocalGraphicsEnvironment();
ge.registerFont(f);

Font dvs = new Font("DejaVu Sans", Font.PLAIN, 20);

Here's an image with plain (dvs) and derived bold (dvs.deriveFont(Font.BOLD)) font.

Test 2

Font f = Font.createFont(Font.TRUETYPE_FONT, new File("dvs.ttf"));
Font fb = Font.createFont(Font.TRUETYPE_FONT, new File("dvsb.ttf"));

GraphicsEnvironment ge = GraphicsEnvironment.getLocalGraphicsEnvironment();
ge.registerFont(f);
ge.registerFont(fb);

Font dvs = new Font("DejaVu Sans", Font.PLAIN, 20);
Font dvsb = new Font("DejaVu Sans", Font.BOLD, 20);

And here's an image with plain (dvs) and truly bold (dvsb) font.

You can confirm that correct file is used by looking at font2DHandle.

I also tested italic and bold italic and both worked as well as Font#createFont(int, InputStream) method.

Above approach works because fonts are mapped by their full name (e.g. Arial, Arial Bold etc.), so as long as your fonts are correctly named you can register multiple members of one family.