Not exactly. The so-called "nuclear option" (as coined by Trent Lott, of all people) is a parliamentary maneuver going back to the early 1800. Without wading too far into the reeds, it's really about doing away with Rule 22 by actually changing the Senate rules. It would kill the filibuster, probably forever.
The Budget Reconciliation process has only been around since 1974, by way of the Congressional Budget Act of 1974. It was amended a couple times, most recently in the 80's with a rule that came to be known as "the Byrd Rule". Byrd didn't like how the process was being used, so he crafted some limits.
"In this case, the "nuclear option" is to appeal to the parliamentarian that the business is appropriate for the reconciliation process"
Kind of. Before the bill come to the floor for a vote, the Senate Parliamentarian will evaluate every provision of the bill, and he has unilateral power to strike whatever provisions he deems not conforming to the Byrd rules. BUT, the President of the Senate - Biden in this instance - has the power to unilateral overrule the Senate Parliamentarian. If the GOP wants to overrule the President of the Senate, they have to come up with 60 votes - of course, that's not going to happen.
I guess if you want to call that "nuclear", then it's nuclear. Certainly, Reconciliation has been used before. But, since its inception in 1974, the President of the Senate HAS NEVER overruled the Senate Parliamentarian. Trent Lott fired the former Parliamentarian because he didn't like the rulings he got, but Dick Cheney refused to intervene, with good reason.
I think the Republican needed it because they could not guarantee they would get a majority vote with Lincoln Chafee, Olympia Snowe, and Susan Collins likely to vote with Democrats. Hence, the maneuver where the parliamentarian rules and the VP overrules.
-PJ