Legally speaking, threats fall under "assault". If I raise my fist to you and step up and punch you, I'll probably be charged with "assault and battery". Where "assault" is the "imminent, credible threat of physical violence" of raising my fist and approaching you, and "battery" is my actually hitting you, (and if I miss or you dodge, that trades n the Battery for a second Assault charge) It's an important distinction because the laws and consequences differ
A threat of physical violence must be credible to be assault. That usually places the bar at "a reasonable person would genuinely fear for their safety as a result of the words and actions".
"Remote threats" are handled a bit differently. They used to go through the US Mail and so were addressed with "mailing threatening communications" (part of 18 U.S. Code) although that requires a lot more investment and consideration. Get out pen and paper, think about what you're going to write, draft the threat, stuffit in an envelope, add a stamp, take it to a mail box. That involves lots of time to reconsider, plus the investment of time, paper, envelope, stamp, and finally time to go mail it. (all while reconsidering and being able to change your mind) Most threats never made it into the mailbox as tempers cooled and emotion gave way to reason. But if it made it all the way through that process, a reasonable person would more easily be able to conclude the threat was credible
Now, it's fifteen seconds of "furiously type a line or two of rage and click Send." And it's handled by the FCC now since it's using an interstate communications network. There's a separate "legal bar" for it to pass, but it's essentially the same thing. The laws are much more recently authored, and so require a bit more since courts now rely more on letter of the law than interpretation by a judge. They're looking for Intent (specifically, are you venting, intimidating, or announcing your intentions), does the message describe a credible threat to the victim, and is it specific about what's being threatened.
"I'm gonna dance a jig the day someone ends you!" - lacks intent
"if I get my hands on you l'm going to launch you into space and watch you suffocate!" - not credible
"you're going to regret the day you pissed me off!" - not specific
(Law Abiding Citizen demonstrates masterful avoidance of legal classifications by careful choice of words)
Although as mentioned above, power and money can press a thumb down on the scales of justice and get an investigation launched regardless of established legal standards.