Confused as to how to terminate a program

I'm confused as to how to terminate programs in bash.
From my understanding, Ctrl + c (command + c for macs) sends the term (terminate) signal to the process directly; it tells the process to please terminate. This allows the program to perform clean-up operations and exit smoothly. You can only seem to run this command from w/n the process....so if you are not running the process yourself, or it is running in another bash instance, you cannot use this command to terminate a process.
However, kill PID# seems to send the same (term) signal....is this the same as Ctrl + c (command + c for macs), but is used when you are outside of the process or the process is open in another bash instance?
Also, am i understanding correctly that the kill command sends the signal to the operating system kernel, which shuts down the process instead of sending the signal to the process itself?
I also read that the kill command actually sends the term signal, so does kill -TERM PID# do the same thing as kill PID#?
kill -KILL PID to me seems to be the syntax to send the kill signal to the OS kernel directly....
Would greatly appreciate if someone could clear this up for me!