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.
Your interpretation makes a mockery of the language used by our Founders. “In all such Cases”, clearly restricts the Cases referred to. If the Founders intended that “In all such Cases” refer to all votes taken, they would have clearly stated that.
“Cases” and “votes” are two entirely different things.
Want more proof? Consider the many voice votes taken that do not have the Yeas and Nays recorded as required by the clause in question (”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”).
For your interpretation to stand, all voice votes are in violation of the Constitution.
You're leaving out the word "such". This is the critical operative word. It's drawing direct reference to votes taken and recorded for overriding presidential Vetos.
Throughout the history of the House, especially in the first half of its existence, plenty of legislation was passed in by voice vote only without individual votes being recorded. The House rules, not the Constitution require in the contemporary House, that a roll call vote is held for some specific kinds of legislation - namely revenue bills and the like. But, like any House rule, this can be amended, if only temporarily. Also, the Speaker is given tremendous latitude with respect to what rules are or are not enforced. As a practical matter, virtually all legislative votes are recorded today, but there's no Constitutional requirement to do so - just the votes to override a Presidential veto.