Do I really need a DB9-RJ45 adapter to connect to a Cisco router console, and if so, why?

Solution 1:

While using a classical RJ45 jack, the connection on your Cisco router really is a RS232 serial connections.

This means that, independently from the physical connector, your PC need to speek the same RS232 protocol. This is why do you need a DB9 (classic serial port connector) to RJ45 converter. The converter does not change the underlying protocol, rather it only provided the correct wirings for the right pins.

On the other hand, your PC's RJ45 connector is not internally connected to a serial port; rather, it is a direct extension of your network card, which speak a completely different protocol (even at the basic electrical layer).