It is better that ten guilty persons escape than that one innocent suffer.
Differing Standards for House of Representatives and the Senate
On the question: Why two standards, namely simple majority vs supermajority (two-thirds)?
The answer lies not so much in the intention or deliberations of America's Founding Fathers. So, instead of seeing it as deliberate decision on the part of the Founding Fathers, the better enquiry is to ask, what is the purpose of each requirement (simple- vs super-majority).
If we adopt this perspective, what is the purpose/meaning of each requirement, we can see that a simple majority is the (constitutional) equivalent of "probable cause" or "reasonable suspicion" standard. To convict, on the other hand, the supermajority (two-thirds) by the Senate is clearly a higher (more difficult) standard. Again, using the constitutional equivalency, Senate cannot convict on mere "probable cause" but the standard of "beyond reasonable doubt".
If I can paraphrase into America's context, it takes a lower standard to suspect and charge - impeachment by Representatives by a simple majority. But it takes a lot more to convict - two-thirds of the Senate.
Legal Doctrines Galore
Legal doctrines being what they are (often also how they are expressed and intrepreted) tend to confuse many. My interest in OP's question is to see if I'm still confused having done the reading (too long ago).
Let's start with what this question is not about. It isn't about the legal doctrine of separation of powers. For which, Hamilton's Federalist Paper #66 answers quite succinctly. Neither is it about when to impeach and what instrument Parliament (or in this case, Congress) should use to remove an officer of the Crown or a federal officer in the case of USA.
There are many doctrines referenced in the Federalist Papers, in general, as would be expected in a debate about institutional powers. But I have not read anything in the discussions on explicitly setting out the standard of proof required to impeach and convict a public official, i.e. OP's enquiry.
So, where did this standard or requirement come from?
Lex ex Consuetudo Parliamenti
Latin for law and custom of Parliament. The common law system is, to put it mildly, highly unsatisfying the deeper one gets to its (historical) roots. This is one of those occasions.
"Lex ex consuetendo Parliament" is not criminal nor civil nor even canon law. It is purely the evolved practices of Westminster Parliament, one that only the High Court of Parliament practices.
From Hatsell's Precedents (Precedents of Proceedings in the House of Commons: Relating to Conference and Impeachment):
“That matters moved in Parliament shall be managed, adjudged, and
discussed, by the course of Parliament; and in no sort by the Law
Civil, or by the Common Law of the Land, used in other lower courts of
Sir Edward Coke says, “As every court of justice hath
laws and customs for its direction, some by the Common Law, some by
the Civil and Canon Law, so the High Court of Parliament suis
propriis legibus et consuetudinibus consistit. It is by the Lex et
Consuetudo Parliamenti, that all weighty matters concerning the Peers
of the Realm, or Commons in Parliament assembled, ought to be
discussed, ad judged, and determined.”
Indeed all the wisest statesmen and greatest lawyers, through a long succession, from
Sir Edward Coke and Mr. Selden, to the Earl of Hardwicke, have, whenever an opportunity has been offered to them, constantly repeated this doctrine.
Since 1760s at the latest, and very likely earlier, before Blackstone's Commentaries on the Laws of England, it is the heritage of common law system of setting higher standards to convict (as opposed to merely charge or indict). This is to err on the side of caution, lest we convict the innocent (Blackstone's ratio).
How did this come to pass that US adopted this doctrine into their Constitution, resulting in differing standards to impeach by the House of Representative on the one hand, and on the other, a higher standard of two-thirds by Senate to convict?
The answer lies in the law and custom of the British Parliament ('Lex ex consuetendo Parliament').
Since I began with a line from Blacksone's Commentaries, I wonder if it's fitting to end with another:
The president is not Gulliver immobilized by 10,000 tiny cords, nor
even a Prometheus chained to a rock of frustration. He is, rather, a
kind of magnificient lion who can roam widely and do great deeds so
long as he does not try to break loose from his broad reservation.