For example, say I am learning how to play Cole Porter songs on piano (let's misbehave;de-lovely;etc) yet I come across other versions of the song, some have a higher level of difficulty and I feel if I play a different version of the song, then I can't say I am 'really playing' the way Cole Porter, really played, and cheating myself of the authenticity of the original. Am I going about this the wrong way? What are your thoughts when you come across other people's arrangement of popular songs especially from top sheet music books and companies like Hal-Leonard etc??
With popular music, many of the published stuff is a poor representation of the composer's original intent. I've seen fake books which were clearly just compiled by someone listening to the music and writing down what they thought they heard. The best approach is to find an original publication or manuscript (there's a whole field of study in how to do this, especially for classical composers.) One can read reviews of various editions of Chopin or Beethoven and find that there are many mistakes.
In the pop (and perhaps the jazz) field, people may only write down a melody line with suggested chords and sometimes lyrics. Sometimes the transcriptions get into trouble by trying to combine the melody with the chords and put a bass line with slash chords. This makes it hard to correct errors (if something transcribe is obviously wrong, the transcriber can use musical knowledge to repair things).
Lyrics are just a bad. Most sources of lyrics on the 'net come from listening, sometimes by people who don't even speak the language.
Your best bet is to check through (for Cole Porter whose music is rather complex for popular music) older sheet music or biographies. You may even find something that was checked over by Cole himself.
Those kinds of arrangements can be very different that the original works.
In fact in can be difficult to even say what the "original" composition is. What is the definitive version of "I Got Rhythm?" It was originally from a musical. Whatever early sheet music version for piano and voice can be found will be an arrangement. It will probably be the work of some editor of the publisher. Sheet music in the early 20th century was a way to mass distribute music before the advent of electronic recordings. It was meant for amateur players at home and not necessarily what the original musician performed.
In the case of musicians who lived in the era of recorded music you can try to find transcriptions of their performances. In some early cases the records are piano rolls.
Other than that I think you would need to look for monographs or autograph documents. The actual handwritten documents from the composer. Take a look at the description of this Cole Porter collection at the Library of Congress.
...considered to be the closest thing to original manuscripts for these songs as exists...
Very loosely related to your actual question: you might want to check out the movie De-Lovely which is about Cole Porter. I've read that Kevin Kline actually sang the songs and played the piano for the filming. Perhaps a source of interpretation for you?
If you want a definition of 'classical' vs. 'popular' music, it might be that in the former the manuscript is urtext, in the latter a recording is.
Cole Porter was writing for Broadway, for Hollywood. Even if he did prepare a piano/voice manuscript, that would rarely have been intended as the definitive version. And, although he played piano, he wasn't known as a performer.
The definitive version of a 'Golden Age' popular song is probably the arrangement used in the movie or show that made it a hit. Which is why it's slightly amusing to see the published version played as if it's Bach's autograph. That arrangement was probably made by a publishing house staffer.
It might be of interest to check an Elton John or Nat King Cole recording against the published copy though. They often DID play piano as a basic part of the definitive recording.