What does "lawful presence" mean?

The meaning of the phrase seems to lie at the heart of the dispute over the legality of the president's deferred action policies regarding two classes of undocumented immigrants. The case - United States v Texas - was argued today before the court. One of the arguments opponents to the executive actions made was that the use of the phrase in the regulations was the same as a creating a whole new immigration law. Supporters seemed to suggest that was not the case and that the policy would be effectively the same whether or not the phrase existed in the original memorandum setting out the policy.

- Click here for it.

Here's language form the brief submitted by the state of Texas detailing their complaint. which ultimately boils down to the president usurping legislative power which properly belongs to Congress:

- Click here for the brief.

The Executive Branch unilaterally created a program—known as DAPA—that will grant “lawful presence” and eligibility for work permits to over four million aliens who are present in this country unlawfully. “Lawful presence,” an immigration classification established by Congress, allows aliens to receive numerous benefits—such as Medicare, Social Security, the Earned Income Tax Credit, and unemployment insurance. And Congress has created a detailed, complex statutory scheme for determining when an alien may lawfully enter and be present in this country. 

The Executive claims the power to ignore these statutes and unilaterally deem lawful the presence of any unauthorized alien it chooses not to remove. Moreover, the Executive asserts that it may do so for mil- 2 lions of aliens without even using conventional noticeand-comment procedures. But the Executive does not dispute that DAPA would be one of the largest changes in immigration policy in our Nation’s history. The President himself described DAPA as “an action to change the law.” Pet. App. 384a; R.69.1 There is no statutory or constitutional authority for such a change. 

- The adminstration's response is here.

ScotusBlog adds the following:

- Click here for it.

At repeated points during the argument, it appeared that the Obama administration might actually be able to salvage its policy as both legal and enforceable (if the lawsuit were allowed to go forward) by simply deleting from the regulations in dispute a two-word phrase: “lawful presence.” It was very clear during the challengers’ argument — by lawyers for the twenty-six states and for the U.S. House of Representatives — that the phrase carries enormous meaning for them, that they see it as the equivalent of a whole new immigration law enacted by the White House, not Congress.

At one point, Justice Anthony M. Kennedy, who could wind up holding a decisive vote, appeared to hint that he agreed that this is what happened, that in this instance, the normal order of government policymaking had been turned “upside down,” as he put it.
The Obama administration’s lawyer sought to defuse that complaint by telling the Court, in response to several questions about the phrase, that it stands for nothing whatsoever, and could simply be deleted without affecting the policy in any way. But, if that is the solution, would it be enough to generate five votes in favor of the policy.