what's the difference between `docker stop` and `docker kill`?

Is it that docker stop attempts to stop the process run inside the container in the correct way, while docker kill will send a kill signal?

Basically yes, the difference is subtle, but outlined in the Command Line reference:

• docker stop: Stop a running container (send SIGTERM, and then SIGKILL after grace period) [...] The main process inside the container will receive SIGTERM, and after a grace period, SIGKILL. [emphasis mine]
• docker kill: Kill a running container (send SIGKILL, or specified signal) [...] The main process inside the container will be sent SIGKILL, or any signal specified with option --signal. [emphasis mine]

So stop attempts to trigger a graceful shutdown by sending the standard POSIX signal SIGTERM, whereas killjust kills the process by default (but also allows to send any other signal):

The SIGTERM signal is sent to a process to request its termination. Unlike the SIGKILL signal, it can be caught and interpreted or ignored by the process. This allows the process to perform nice termination releasing resources and saving state if appropriate. It should be noted that SIGINT is nearly identical to SIGTERM.

While not enforced in anyway, processes are generally expected to handle SIGTERM gracefully and do the right thing depending on their responsibilities - this can easily fail due to the graceful shutdown attempt taking longer than the grace period though, which is something to consider if data integrity is paramount (e.g. for databases); see e.g. Major Hayden's SIGTERM vs. SIGKILL for a more detailed explanation:

The application can determine what it wants to do once a SIGTERM is received. While most applications will clean up their resources and stop, some may not. An application may be configured to do something completely different when a SIGTERM is received. Also, if the application is in a bad state, such as waiting for disk I/O, it may not be able to act on the signal that was sent.

docker kill will stop the main entrypoint process/program abruptly

docker stop will try to stop it gracefully (will ask politely :P)

in both cases the filesystem changes will be persisted (at the time of stop or kill) so if you docker start <container> then it will continue from there.

And addition to the answers added earlier

running docker events after docker stop shows events

• kill (signal 15): where signal 15 = SIGTERM
• die
• stop

running docker events after docker kill shows events

• kill (signal 9): where signal 9 = SIGKILL
• Die (exit Code 137)

docker stop has a timeout before killing the process. The default is 10 seconds.

This table has even more details.