What is the difference between a "Self-Radicalized" terrorist, and one that has been "Radicalized" by some other means?

The ends are the same. The difference is, a “self-radicalized” terrorist is one whose sense of injustice come from observation and not from being oppressed or harmed. An otherwise radicalized terrorist is one who sees a terroristic struggle as the only means fo fighting the egregious injustice on their life.

A kid who grew up comfortably in Boston or in Queens or Lockport NY who identifies with oppressed people in other countries or feels people are bullied by the US Government, and acts terroristically, is self radicalized. A kid who grows up in Mosul and has seen repeated atrocities by various armies and factions, is radicalized as a way to address the violence in his everyday life.

But why bother with the distinction? Especially after the mass murder has already taken place.

The distinction is merely whether someone else was actively involved in forming your thoughts and actions. It can help to determine whether others are culpable in your actions. If we are to fight terrorism, we need to understand how it develops.

This is just a personal distinction but from my point of view a ‘Self-radicalized” person is someone who was looking for an excuse to do what they wanted to do in the first place. They went searching and found something that corroborated their own thought and ideas and channeled them in a direction they were heading in the first place albeit with a more violent bent.
Someone who was radicalized by someone or something else may or may not have been actively seeking radicalization but were open to it when presented with the idea or perhaps were malleable enough emotionally to accept the lessons given as the right ones.

Why bother with the distinction? I am not sure it makes a difference except perhaps to justify going after others.

Seems to me like an attempt to further the Orwellian decline of language accepted by the public at large. First they get us to dehumanize people by calling them terrorists. Then they get us to buy a “War on Terror”, as if that made any sense or could ever end. We also had an “Axis of Evil” and “Patriot Acts” – that seemed to fly with many people, getting people behind tossing our rights and forming secret courts, and a new department of Homeland Security (that’s practically what Gestapo means in 1940s German, Geheime Staatspolizei – homeland state police, by the way), supposedly in order to defend our rights because “they hate our freedom and our way of life” (LOL), and of course invasion of Iraq, drone strikes, etc. The word “radical” is used as another way to shift our symbolic thinking to an “us versus them” mindset. Pretty much anyone speaking the truth against the mainstream corporate machine is also termed “radical” or “extremist”. “Radicalization” is used in a way that makes it sound a bit like a disease or zombie outbreak, to further stoke fear and dehumanization and so foster acceptance of suspending usual laws and restraint in favor of taking action to stop those scary radicalized terrorists… and perhaps distract from noticing how ISIS and Osama were to some degree made what they are by the US, and the other reasons why these people are choosing violence and martyrdom.

By pretending it’s meaningful, important and correct to frame the question (of why people are being violent in these ways) as “are they self-radicalized or not?”, it avoids other questions and makes it sound like the official approved mindset is a matter of significant fact, and that that’s all there is to it, and that thinking need not go further. It’s also a way to point fingers at the next target you’d like to invade or attack or threaten. If you can get people to buy that the bad guys were radicalized by your next target of choice, you can try to sell that as a reason to fear them and so strike first. e.g. Part of the argument to invade Iraq was about them having “terrorist training camps”.

The self-radicalised ones believe they “used their own head” and came to where they are with their own effort and intellect, while the other ones are “manipulated by others” into terrorism.

At the very end, it makes no difference anyways.

GA @Jaxk