If you entered "a", then RPicture#ItemIndex# is set to ""a"" in the bang from Command1 of [InputPicture], from which the outer quotes are stripped off, so the variable is actually "a" - same for writing it to file, so all good till now. However, it seems you only update the [BDE] meter in Command2 of [InputPicture], while the RPicture... variables are also used as image paths in the [IcoR...] and [NowPlayCover] meters. Maybe updating the latter ones as well will do the job (I didn't test it).
I agree with your explanation.
If I enter the example text "Short Text", it is saved under the variable RPicture1="Short Text". (yeah, correctly)
Rainmeter, or rather the text field, resolves this text to Short Text (without ""), which is correct according to your explanation and my understanding.
But I think Rainmeter only resolves the "" correctly after a refresh, but I could be wrong. Here's a small demonstration.
Klick to Play the GIF
Statistics: Posted by Rooky_89 — Yesterday, 1:10 pm