There's a good explanation of the history of these charts at https://en.wikipedia.org/wiki/Billboard_Hot_100#History.

Although I don't know the pre-Hot 100 situation that well, my thinking has long been that if the Hot 100 was basically just a renaming of the Top 100 chart, then the Top 100 would be considered by chart experts to be essentially the same chart -- as is the case with other Billboard charts over the years that were simply renamed. However, the Wikipedia article has a footnote citing Billboard itself as stating "The main chart was Best Sellers in Stores, and that’s the list Billboard uses as THE pre-Hot 100 chart" -- not the Top 100. Here's another item from Billboard on this subject. 

THE DIFFERENCE BETWEEN ‘HOT’ AND ‘TOP’
Dear Fred:
Prior to the formation of the Hot 100 in August 1958, there were four Billboard charts: Best Sellers in Stores (similar to the present day sales chart), Most Played by Disc Jockeys (similar to the present day airplay chart), Most Played in Jukeboxes (obsolete today) and the Top 100. What was the Top 100? If it was a combination of the sales and airplay charts, what made it so different from the Hot 100 which began in 1958?
Thank you,
Andy Ray
Indianapolis, Indiana
Andy46032@inct.net

Dear Andy,
What I know about the Top 100 comes from reading issues of Billboard magazine from the time period. As I understand it, it wasn’t based on one-week stats, but was an average over a number of weeks. What really disqualifies the chart from being counted as the official Billboard singles chart is that it was published with a disclaimer warning retailers and juke box operators not to use it when deciding which records to order, but to use the more reliable charts such as Best Sellers in Stores.
Another difference: the Top 100 had a very brief history; the Hot 100 has stood the test of time as being the most authoritative singles chart in the United States, having just celebrated its 48th anniversary.

I think that your description is accurate, and I also think that it would be helpful if Billboard had an official statement about all of this on their site given the importance of the Hot 100. In addition, I do think that there's every reason to think that "Tequila" would have reached #1 on the Hot 100 if it had existed in 1958. However, the situation with "The Happy Organ" is quite remarkable in any event ... I certainly didn't know that it was originally supposed to have a vocal, and that's particularly interesting to me given that I think it works very well as an instrumental.

aflem wrote:

I think if you re-read the story, you'll see that "The Happy Organ" never had lyrics. It says Cortez was originally supposed to do a separate song called "The Cat and The Dog," but due to voice issues, started fooling around on the organ, presumably on an altered version of "Shortin' Bread." This instrumental was the result. 

The article had led to me looking up Cortez and "The Happy Organ" on Wikipedia, because there were aspects of it that I found to be a bit problematic. The info at https://en.wikipedia.org/wiki/The_Happy_Organ states that:

"The Happy Organ" originally featured lyrics and was intended to be sung accompanied by a piano and an organ. Cortez recorded a vocal for it, but was unhappy with the result. He spotted a Hammond B3 organ in the studio and decided to play the song's melody on it. 

But yes, the info in the article would indicate that it was just "The Cat and The Dog" that was supposed to have a vocal. However, I wondered about the comment at the end of the article that "The Happy Organ" didn't lead to a string of hit records for him, given that he reached #10 with "Rinky Dink" in 1962 -- which it references, but for some reason as having been turned down when Cortez offered it as a B side to "The Happy Organ". Anyway, given that it was also a significant hit for Cortez, I wasn't surprised to see in his bio at https://en.wikipedia.org/wiki/Dave_%22Baby%22_Cortez that he did have six more singles that made the Hot 100 -- in addition to recording numerous albums during the '60s and early '70s. So I think that "The Happy Organ" actually did lead to a pretty significant career for Cortez.