STM32F1/F4: Risks of driving a LED from PC13
The datasheet is very strict about this, the pin must not source current.

And Blue Pill is not wrong, the pin sinks current to turn LED on.

Sourcing and sinking in this case just mean direction of current relative to something. IO pin that sources current means current flows out of MCU (it comes from positive supply), and IO pin that sinks current means current flows in to MCU (it goes to ground).

Inside the MCU there is an area called Backup Domain that can be powered via the VBAT battery supply pin even when main supply is turned off, so that real-time clock can keep time and have some SRAM that retains contents during main power loss. Also the IO pins PC13/PC14/PC15 that are related to this Backup Domain must be powered from Backup Domain. There is a switch or supply voltage multiplexer that selects VBAT supply to backup domain when there is no VCC supply available. This supply multiplexer or switch can pass, or in datasheet terms, sink the few microamps that normally passes through it, but can't pass much current or it might break or the voltage drops too much for the backup domain to work properly.

It does not mean there is a constant current sink anywhere. It just means the internal power switch can handle up to 3mA safely in all situations, and backup domain and the power switch itself will stay operational when these three special weak IO pins are used within the limits given.

So, as the switch can provide limited amount of current to Backup Domain, that is why the datasheet says the three IO pins must not be used as a current source. It just means there can be no load that draws current out of the pin. It is acceptable to drive e.g. few CMOS gate inpus, as they draw no DC current and the load is mainly capacitive. Since fast transitions to drive capacitive loads need larger currents than slow transitions, that is why the datasheet says to limit the output pin drive strength to the weakest 2MHz setting, and to limit the capacitive load to a maximum of 30pF.