1. Create an empty file when the application launches
2. When the application terminates, write the reason for termination into the file.
3. When the app launches again, read the file. If it was empty, then you were SIGKILL'd, or you crashed.
"Might have been killed, might have crashed" is not helpful, and the inability to know is the very issue at hand here. You can't know whether your application crashed, the user forcefully killed it or you fell afoul a background killer. What now, do you start spamming the user anyway?
You might be able to write the word "bad" in a file, and then clear the file if your app closes nicely. If you start and find the word "bad" in the file, then you know you were killed. A slight variation of what you suggested.