1. "Al-Sham" is a word often used for Syria, and more specifically for Damascus. "Bilad al Sham", on the other hand, is Levant or Greater Syria.
2. When the leader of Jabhat al-Nusra spoke about their move from Iraq to Syria to form Jabhat al-Nusra, he spoke of moving to "al-Sham".
3. When ISIS was formed, they certainly didn't mean the group would operate in all of Greater Syria or Levant. It was only recently that they announced they would open a branch in Lebanon.
4. Often when there are Arabic words whose translation into English is disputed, it's better (academically speaking) to use the Arabic word and explain what it is – in this case, the Islamic State of Iraq and al-Sham (still ISIS).
5. If we concede again that "al-Sham" means not only Syria, then there is a name for that: Greater Syria. When we use the older term "Levant", that should be used alongside the older name "Mesopotamia" for Iraq. When you use modern "Iraq", use the modern term "Greater Syria" -- in that case, it's the Islamic State of Iraq and Greater Syria (still ISIS).