Saving someone else's video, where the line actually is
Downloading is not the risky part. What you do with the file afterwards is where people get themselves into trouble.
The file is a copy, the rights stay with the creator
Pressing save does not transfer anything. The person who made the video keeps every right they had, exactly as if you had never touched it.
That is worth holding onto because it answers most questions on its own. If an action would need their permission when the file lived on their phone, it needs their permission now.
Generally fine
Keeping a copy of a clip to watch offline on a flight or a commute.
Saving your own uploads back off a platform, which is the most common reason people reach for a downloader at all.
Building a private reference folder while you study how a creator edits.
Generally not fine
Reposting a clip as your own, cropped, sped up, or with the watermark removed to hide where it came from.
Running someone else's footage in an advert or any monetised channel without a licence.
Collecting videos of a private individual who did not intend a wide audience, even where the post is technically public.
Asking is cheaper than arguing
Most creators say yes to a polite request with credit attached, because a repost with a tag brings them followers. A repost without one brings them nothing and a takedown form.
If you cannot reach them, take the idea rather than the file. Ideas are not owned, and your own version will fit your audience better anyway.