The easiest way to bypass this semantic confusion is to explicitly say that it's about always having the option of continuing to be alive, rather than what people usually mean by immortality.

1) Death can still happen from any number of causes - tornadoes, for example.

2) That may bypass some of the most conscious semantic confusion, in the same way declaring that whenever you said any number, you always meant that number minus seven would clear up some confusion (if you did that). There is a better way.

3) It's probably not true.