Your Art. I section 2 reference includes restrictions/qualifications. The others you list don't. Thus the latter all refer to the same group, the people of the United States. The Former only to qualified voters, except to the extent that the People of the Several States are represented in that voting by the qualified electors.
I use it because it, constitutionally, defines "the people" as those who select House members.
Since the individual states had different requirements for the electors of state senators and the electors of state representatives, the second half of Article I, Section 2 clarifies which electors they're referring to.
It should be clear to anyone reading that exactly who "the people" were.
"The others you list don't. Thus the latter all refer to the same group, the people of the United States.'
Why would the Founders do that? Why would they have "the people" mean one group in Article I, Section 2, and something different elsewhere?
Look at the constitution. The Founder were quite specific. When they meant citizens they wrote "citizens". When they meant an individual or all individuals (e.g., the 5th amendment) they wrote "person" or "persons". And when they meant the enfranchised body politic, they wrote "the people".
You say the latter refers to "the people of the United States". I have no idea what you mean. Are you being intentionally vague? Who are, specifically, "the people of the United States"? All individuals, including foreigners? Just citizens? Just adult citizens? Did it include women and children and non-whites in 1792? Does it today?
I mean, nice theory, but it crumbles when examined.