The order refers in sec. 3(c) to "immigrant and nonimmigrant entry into the United States of aliens from countries referred to in section 217(a)(12) of the INA, 8 U.S.C. 1187(a)(12)". That section specifically names Iraq and Syria, plus "points to" potential areas of concern, as identified by the Sec'y of State and Sec'y of Homeland Security. The full list extends to Iran, Libya, Somalia, Sudan, and Yemen, and is set forth at 81 FR 39680.
The cited authority is 8 USC 1182(f), a section about inadmissible aliens, which says
Whenever the President finds that the entry of any aliens or of any
class of aliens into the United States would be detrimental to the
interests of the United States, he may by proclamation, and for such
period as he shall deem necessary, suspend the entry of all aliens or
any class of aliens as immigrants or nonimmigrants, or impose on the
entry of aliens any restrictions he may deem to be appropriate.
Thus he has authority to exclude permanent residents from re-entry (they are aliens, and re-entry is a subcase of entry), and has done so for 90 days. In the case of Syrian nationals, under sec 5(c) of the order, they are excluded until the order is changed. Sec. 5(c) of the order specifically declares
Pursuant to section 212(f) of the INA, 8 U.S.C. 1182(f), I hereby
proclaim that the entry of nationals of Syria as refugees is
detrimental to the interests of the United States and thus suspend any
such entry until such time as I have determined that sufficient
changes have been made to the USRAP to ensure that admission of Syrian
refugees is consistent with the national interest.
The long-term ban on Syrians also specifies "as refugees", so "such entry" might not be interpreted to refer to entry of refugee or non-refugee Syrian nationals who are permanent residents. The 90-day ban in 3(c) ("immigrant and nonimmigrant entry into the United States of aliens") does not include any further restrictors, apart from the various government-official visas that are exempted, so there would be no basis in the order for excluding permanent residents. Whether this will be strictly enforced is a separate question, though in the period when the order was active (a day), it was applied to permanent residents. Sec. 3(g) allows exceptions "on a case-by-case basis, and when in the national interest". A recent (Saturday the 28th) White House briefing says that green card holders "will need a case by case waiver to return to the United States".
The ban includes immigrants and non-immigrants, so it doesn't particularly matter how "immigrant" is defined – what matters is what an alien is. Still, "immigrant" is defined in 8 USC 1101 (15) as any alien, except a bunch of categories such as diplomats, tourists, business visitors, people in transit, students, and fashion models and similar, working temporarily here. Any permanent resident has an immigrant visa (but at any rate is an alien, and is either an immigrant or a non-immigrant).
Returning to the words of the order, neither 3(c) nor 5(3) limit the ban to issuance of visas, it refers to entry. Not all entry by foreign national requires a visa. The word "visa" is mentioned, so it is possible to construe that mention as contextually restricting the ban to visas, however 3(a) which is entitled "Suspension of Issuance of Visas and Other Immigration Benefits to Nationals of Countries of Particular Concern" orders the review "to adjudicate any visa, admission, or other benefit under the INA" which indicates that there is no such limitation.
8 USC 1101(a)(13)(C) does say that a permanent resident is not "regarded as seeking an admission into the United States". Admission itself is defined in 8 USC 1101(a)(13)(A), when applied to an alien, as "the lawful entry of the alien into the United States after inspection and authorization by an immigration officer". While a permanent resident is not seeking admission (and is not applying for admission), it would still be consistent with the law to forbid admission (legal entry). If you read 8 USC 1101(a)(13)(C) as granting permanent residents widest latitude to enter without restriction, then that has the absurd consequence that permanent residents are subject to no scrutiny at all. The absurdity is avoided if "seeing admission" is a step prior to and is distinct from "being admitted" i.e. "lawfully entering".
