kill -INT vs kill -TERM
The only difference in the response is up to the developer. If the developer wants the application to respond to SIGTERM differently than to SIGINT, then different handlers will be registered. If you want to stop a background process gracefully, you would typically send SIGTERM. If you are developing an application, you should respond to SIGTERM by exiting gracefully. SIGINT is often handled the same way, but not always. For example, it is often convenient to respond to SIGINT by reporting status or partial computation. This makes it easy for the user running the application on a terminal to get partial results, but slightly more difficult to terminate the program since it generally requires the user to open another shell and send a SIGTERM via kill. In other words, it depends on the application but the convention is to respond to SIGTERM by shutting down gracefully, the default action for both signals is termination, and most applications respond to SIGINT by stopping gracefully.
If I wanted to stop some background process gracefully, which of these should I use?
The unix list of signals date back to the time when computers had serial terminals and modems, which is where the concept of a controlling terminal originates. When a modem drops the carrier, the line is hung up.
SIGHUP(1) therefore would indicate a loss of connection, forcing programs to exit or restart. For daemons like syslogd and sshd, processes without a terminal connection that are supposed to keep running, SIGHUP is typically the signal used to restart or reset.
SIGINT(2) and SIGQUIT(3) are literally "interrupt" or "quit", "from keyboard", giving the user immediate control if a program would go haywire. With a physical character based terminal this would be theonly way to stop a program!
SIGTERM(15) is not related to any terminal handling, and can only be sent from another process. This would be the conventional signal to send to a background process.