The one time my neighbors called for help (their car wouldn't start and they needed to go to urgent care), they txt'ed first, then called. Though I would have picked up when they called (and it would have gotten through my do not disturb settings) because their number is in my contacts.
Not saying that your neighbor would call you, but that the emergency services would call you on your neighbours behalf (and it will not be from a number you recognize).
I've never heard of that happening, but yeah, if the neighbor calls 911, and then EMS calls me for some reason, I'm not going to answer since 99% of the time it's a spammer.
I have lost track of how many times I have told people: just because you hear a phone ringing on your end, doesn't mean it's ringing on my end. (Verizon US, and yes, you as a caller will hear ringing while my phone sits silent.) Leave a voicemail. It can be short, it can be almost devoid of meaning, but if you send it, I will at least know that you called. If you don't? No record at all. I was out of range, you left nothing to be tracked by...