There's more involved here than words. "Islamist terrorism" is what you want to say when you talk about ISIS and al-Qaeda. They are properly "Islamist" because "Islamism" and "Salafism" are the names in English for the offshoots from worldwide Islam that spawned their movements.
You can tell right off that the Islamists have departed Islam. The al-Qaeda crews. They tossed out two of the Five Pillars as laid down by Mohammad: Charity and the Hajj (pilgrimages to Mecca.) Their followers drop them both.
The Salafists keep the Five Pillars, but they join Islamists by rejecting Mohammad's fundamental statements on the religious status of Jews and Christians: that all are blessed and, if good people, will be with Allah in Paradise:
Quran, Sura 5,69: “Surely they that believe, and those of Jewry, and the Christians, and those Sabeaans, whoso believes in God and the Last Day, and works righteousness – their wage waits them with their Lord, and no fear shall be on them, neither shall they sorrow.”
The Salafists started it by altering the Quran. The Salafis added hateful words and whole clauses to every applicable verse.
Imagine a Christian bible rewritten by Joseph Goebbels. Same difference.
Part of how they've gotten away with it scheme is that the ancient Arabic of the 7th Century Quran is now a Dead Language. No one speaks it in ordinary life. Ancient Arabic is as far from the modern dialects of Arabic as Roman Latin is removed from French and Italian.
People do memorize the original Quran phonetically. Unfortunately very, very few of them understand what they are memorizing. At the very worst of this perversion, both the Islamists and Salafis have taken to quoting a work from an Egyptian, Sayyid Qutb, "Fi Zilal al-Quran" in place of using verses from the Quran.
Qutb (1906-1966) traveled to Chicago in the United States as a student during the Truman years. He came back to Egypt hating everything American, modern, or scientific. And hating Jews, particularly. His obsessions make quite a list. Qutb's rants live on through al-Qaeda and ISIS as motivators for their trademark pathological behaviors. He is cited repeatedly in connection with such as murdering tourists in Egypt, the ISIS beheadings, trying to exterminate the Yazidis, and pursuing sectarian war all over MENA.
Qutb is their poet of hatred.
Sayyid Qutb exalted No Mercy predations on ordinary people which makes him practically unique among world writers. His rival in the West is the Marquis de Sade, who was an ineffectual dilettante compared to what Sayyid Qutb has produced. Qutb glorified what the most of the world categorize as psychopathic behaviors -- formed up as an extreme xenophobia -- which he aimed at group after group of human beings.
The Marquis de Sade died an old man, in bed. Qutb was hanged for his role in a bloody assassination attempt and rebellion out of the Muslim Brotherhood movement in egypt.
(Nasser's secular nationalism ran opposite to the extremists in Muslim Brotherhood. However, Nasser still tried to follow theories of pluralism by offering Qutb a post in government leading Egyptian education. Perhaps Nasser did not understand just how crazy this guy was. By 1964 Qutb got himself involved recruiting for a plot to kill every one of the top government figures in Egypt. What could go wrong with that? You've seen similar patterns replaying recently in Egypt and in Syria.)
When al-Qaeda in Iraq (AQI) rebranded themselves as Islamic State in the Levant, they were trying to reframe where they fall in relation to mainstream Muslims.
In fact, they are not people of the original Quran. Not at all. They follow the Salafi's rewrites of the Quran. They follow Sayyid Qutb, not Mohammad. The image they have of Mohammad is more a cartoon creation, properly a figure in their special Cult of Mohammad.
All in service of hatred.
There's nothing "Islamic" about it. What ISIS and al-Qaeda need to fill their ranks are young men who want to kill somebody. That's the job. Their recruits come over to Syria or go into terror cells. They learn how to kill people. They go out and that is exactly what they do -- they kill people. There is nothing productive about it.
We saw the same kind of behaviors from an army of Serbians during the Bosnian War. They claimed to be Christians. They got control of Muslim areas such as Srebrenica. There they Killed 8,000 Muslim men in one massacre and dumping their bodies in mass graves. This was followed by holiday celebrations and much consumption of beer and hard liquor.
Those Serbs got off on it -- high time in the old town !!
Ratko Mladic and Sayyed Qutb deserve, if I may say so, their own shared cage in Hell.
Anyway, please do not blame ordinary Muslims for what the Islamists and Salafists and Qutbists and Muslim Brotherhood in Egypt and Syria have wrought. The ordinary Muslims and straight translations of the Quran preach mercy, charity, brotherhood with other Peoples of the Book, tolerance of others, and forgiveness of all sins at least once a year.
"Ordinary" equals "Islamic."
And "Islamic" belief and practice are anything but what the Islamists holes in ISIS and al-Qaeda are doing.
As to why Republicans and Israel's Likudniks insist that ISIS and al-Qaeda are part of mainstream Islam, I think you have to Follow The Money. They pump up bigotry and a Qutb-like xenophobia but against Muslims. That's politics.
The end point of pumping bigotry is that it pumps up Perpetual War.
There's money to be made. There's political power. And all the right wing American and Israeli politicians need to get it rolling is an enemy.
(Damned communists! They failed.)
So instead of targeting ISIS and al-Qaeda, they target the whole of the world's 1,600,000,000 Muslims. You see Netanyahu, McCain and Graham out front as talking heads. Giuliani and Peter King playing the fool. You see their talking points about "extremist Islamic terrorism," all based on the "Islamic State" tag.
Properly considered, their crap is nonsense. There is no reason to go along with ISIS propaganda. "Islamic" Islam is not terrorism, ever.
You want to blame somebody, it is the Saudis who have Salafism as their state religion. It's a cult and they do it.
They have financed extremists all over and pumped hundreds of millions of dollars into the United States.
Black Gold Rules.
Have a look at Carlyle Group and the investment history. They had G.H.W. Bush and James Baker as "employees" when they cashed in with their massive investments in U.S. military contractor firms made at the time of the 9/11 attacks. Not entirely wonderful for ordinary Americans.
Perpetual War is good for the very rich, which applies everywhere. Setting up American bigotry against 1,600,000,000 Muslims instead of against 40,000 or 50,000 nut cases serves the cause.
So expect to hear "extremist Islamic terrorists" over and over.
Typical for Republicans. They're not just wrong. They're xssholes.
Note: there's another meaning for "Islamist." That is a western expert on the Middle East. Long beard, a pipe, many books in Arabic, preferably an Oxford Don if you please.