I hate to say it, but I think you are right.
Conservatives, including Mark Levin, are quoting Article I, Section VII, Clause II, but that portion refers to votes over-riding presidential vetoes, not votes for passage of legislation. (It begins with, "But in all such Cases . . . ", meaning it only refers to specific cases, not all cases. The description of the cases it refers to precedes that clause - voting to over-ride a presidential veto.)
Even so, if the Dims adopt that rule, it will open up a thermonuclear war between the parties, and will probably produce Republican majorities in both houses in November. IMO.
That's how I read it. Levin probably knows better. This is just politics, where the facts are secondary.
Every Bill which shall have passed the House of Representatives and the Senate, shall, before it become a Law, be presented to the President of the United States; If he approve he shall sign it, but if not he shall return it, with his Objections to that House in which it shall have originated, who shall enter the Objections at large on their Journal, and proceed to reconsider it. If after such Reconsideration two thirds of that House shall agree to pass the Bill, it shall be sent, together with the Objections, to the other House, by which it shall likewise be reconsidered, and if approved by two thirds of that House, it shall become a Law. But in all such Cases the Votes of both Houses shall be determined by Yeas and Nays, and the Names of the Persons voting for and against the Bill shall be entered on the Journal of each House respectively, If any Bill shall not be returned by the President within ten Days (Sundays excepted) after it shall have been presented to him, the Same shall be a Law, in like Manner as if he had signed it, unless the Congress by their Adjournament [sic from online PDF source] prevent its Return, in which Case it shall not be a Law.
You appear to have a novel understanding of the English language. This paragraph lists three distinct specific instances where a bill must be "passed" by one house or the other (or both).
First, the initial passage. Second, on veto, in the originating house. Third, subsequent to veto and override in the originating house, in the non-originating house.
Then the phrase "But in all such Cases" enters the stadium, which clearly refers to the initial passage, and to the post-veto override alike, unless the definition of "all" has become merely "some".
From the language, it is clear to me that the recorded vote applies to initial passage just the same as override.
'All such Cases' refers clearly to 'Bills' generally, not just to bills vetoed by the President. For 'Cases' to mean that I:7 applies ONLY to veto overrides, one must make the argument that one procedure applies when a bill is being voted upon initially AND A DIFFERENT ONE APPLIES in the case of veto overrides. Curiously, this former mysterious procedure is not described by the Constitution, so by extension one might in the same vein reasonably (cough, choke...) make a 'valid' argument that 'original intent' here means, say, flipping a coin or drawing lots to determine the passage or failure of a bill. A more bogus argument would be difficult indeed to construct.
Also therefore, one would have to make the argument that the House and/or the Senate can, the Constitution notwithstanding, present a finished bill to the President without having to take a vote at all.
Let's see someone make EITHER of those arguments with a straight face!