I can agree with all of that.The "Spot forged text" entry under Calligrapher's Supplies supplies likely refers to spotting forgery directly related to works that specifically feature calligraphy. As per the text for the tool:
My inference is that forging or spotting forgeries of general documents and text aren't covered by this proficiency, but it can apply to mimicking a signature or spotting forged calligraphy within a map, artwork, book, etc.
The Forgery Kit, IMO, would be applicable to mimicking basic handwritten texts, emulating writing styles or legalese, and being able to spot forgeries within those parameters.
I'd allow the two tool proficiencies to be used in conjunction to create very convincing signed documents. It may even be a required step if the signature is well known to the person examining the forgery.
Also of note is that the "Quick Fake" text of the Forgery Kit indicates that an Intelligence (Investigation) check may be used by anyone to spot one of these forgeries, implying that the same skill could be used in general to spot any forgery.
Similarly, the PHB has "Forge a document" listed as a possible task for Intelligence (Investigation). I believe this reinforces the above implication that the skill can also be used to spot forgeries.
So basically we can either grant advantage to Intelligence (Investigation) checks when proficient with the related tool, or perhaps a player proficient in only the tool could use their tool proficiency instead (without advantage).
As for "Spot forged text" in Calligrapher's Tools, I'd argue that it is already covered in this context by "Identify writer of nonmagical script"; i.e. you would not necessarily be able to tell if a fake document is fake, but you could determine if it was written by the same person as another document if you had one for comparison.
Then there is the overlap between "Forge a signature" (Calligrapher's Tools) and "Mimic handwriting" (Forgery Kit). I feel like these should perhaps be switched, but they basically boil down to the same meaning.