what is a 'demon'? (spoilers)

I agree. Even if these "victims" really had done the "evil" things implied, I don't see any connection with something supernatural.

From Wikipedia...

The contemporary Roman Catholic Church unequivocally teaches that angels and demons are real beings rather than just symbolic devices. The Catholic Church has a cadre of officially sanctioned exorcists which perform many exorcisms each year.

Here's text of dialogue from the movie which I found on the internet, which I hope is accurate:

"Dad: The end of the world is coming. It's near. The angel showed me. There are demons among us. The devil has released them for the final battle. It's being fought right now. But nobody knows it except us and others like us."

OP, I don't think he's talking about sinners. He's talking about something else, and it's largely up to the audience to decide what their nature is, apart from the little evidence we are given.

I think they are inhuman creatures, who can be killed not merely as punishment for what they have done, but simply for what they are. Granted, we are not given a lot of evidence that these "demons" have supernatural qualities, but that choice by the filmmakers makes it easier for them to add the twist ending.

If you can accept that Paxton's character is actually being guided by God, he would not need special weapons to hunt and kill the demons if they were just human sinners. As unimpressive as the pipe and the ax seem to be, those were specific weapons God chose for him to use. It is implied that they have special demon-killing power, but I guess you could decide he has to use them just because God told him to.

Also, the father implies that God has told him that Fenton is a demon. Fenton certainly wouldn't make much of an evil-doer at that point compared to the other demons that have been killed. So God has told Paxton's character not about what Fenton has done, but what Fenton is.

The demons we are shown in the movie just seem to want to pass themselves of as human and get away with doing evil, and have no great powers, at least in the face of God's chosen demon-hunters. And it would seem, if Fenton is indeed a demon, that either people sometimes give birth to demons who can pass themselves off as human, or that demons manage to replace existing humans and take over their lives.

If you think there is any doubt there about the nature of "demons," and therefore about the morality of the characters' demon-hunting mission, any such doubt in the characters is eradicated by their religious faith. I think you're welcome to find that unsettling.

They go around and spout "demon" but what they really seem to mean is "sinner".

To claim that a young boy (Fenton) is a demon makes no sense in the context of this supernatural story (even the FBI agent at the end who is a demon seemed surprised that he believed this stuff). However, in the context of a drama (without supernatural elements) it doesn't matter what they call them (sinner/devil(/demon), cause there are no such thing as demons. It's just a word for these "good guys" to filter bad from good humans in their POV