In order to find what USART interface to use you usually need a PCB schematic, which will show what UART pins of MCU are connected to ST-Link.
Normally, you need to download board schematic for your disco, for example:
https://www.st.com/en/evaluation-tools/stm32f4discovery.html#cad-resources
In board schematic you find ST-Link, find its TX/RX pins and check where they go to MCU. Usually they're connected to MCU.
Luckily for you, the document you provided is enough without going into board schematic at all (but I still recommend that you download it, it's nice to have it around sometimes), on page 15 of the document you provided you can see that ST-Link UART is not connected by default, but you also see what pins you need to connect. And it will become USART2 peripheral. Yeah you'll have to connect them with wires.
Alternatively, nothing stops you from connecting any other UART pins without soldering if UART pins of MCU are exposed on Pin Headers. This you will actually have to figure out with board schematic. Remember, that multiple pins can have same USART peripheral functionality (some USART3_RX can be on multiple MCU pins), so check pin functions carefully. Also, you probably know it, but it never hurts to remind that RX must be connected to TX and vice versa (although STM32 can have TX/RX pins swapped).
As for why things are initialized or not by Cube, it doesn't always initialize correct peripherals. You need to check what it activates by default. Usually it's right, but every now and then you need to change something, so don't be surprised.
