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. People forget that ISIS was a merger between Islamic State of Iraq and "Jabhat al-Nusra for the Support of the People of Sham". After a year and two months from creating Jabhat al-Nusra in Syria, Al-Baghdadi unilaterally announced a merger between his group and Jabhat al-Nusra, in other words between the group that operates in Iraq and the group that operates in SYRIA. Jabhat al-Nusra did not mean, and does not say, it is a group for supporting the people of all the Levant but the people of Syria. So the initial merger was Iraq & Syria, not Iraq and the Levant. It was later that the name of the new merger "broadened" to include other areas, as ISIS announced a chapter in Lebanon and more recently in Jordan. So, in short, this is less about the meaning of "al-Sham" and more about that fact.
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).
(I added a few lines to #3, see the ones in italics, in response to the reader comments below)