"pi" is what's known as an "indeterminate symbol."

Oh, come on! That's horrible mis-information, even for you! (Assuming you are a certain individual who posts anonymously because they cannot figure out how to log in despite having been here for years, and cannot keep their promise to stay away whether logged in or anonymous. If you are not that individual, then I'm sorry... but your misinformation is as bad as theirs, in either case.)

An "Indeterminate symbol" is "a symbol that is treated as a variable, does not stand for anything else except itself, and is often used as a placeholder in objects...". That is, it's a placeholder with no specific value -- the traditional x of beginner-algebra books. pi and e, on the other hand, are explicit, well-defined, mathematically understood constants which have a single specific (but admittedly transcendental) value, with procedures that can get it to any precision desired.

Your second sentence,

Its value is truly symbolic, and therefore pure, even though it can never be exactly expressed as any rational number.

It is not truly symbolic. It has a well-known, well-defined value, which is the opposite of "symbolic". It's transcendental, not unknown. But you are correct, transcendentals can never be exactly expressed as any rational number, because they aren't any rational number, they are irrational and transcendental. (edit: added critique of second sentence)

Your third sentence is actually not too bad, so I don't have anything to critique in it.

Your fourth sentence,

Yet, calculus-based equations can use it as-is.

... sets up a false dichotomy with the "yet". You can use those constants in calculus-based equations -- or any other mathematical construct -- and it will work fine because they are well-understood and well-defined, not as a "yet" or "in spite of" their lack of definition. The fact that even a 32bit or 64bit floating-point number implementation of those constants can come so close to the actual value, and the results can stay so close to the real values of the calculations, is a testament to, not an indictment of, the awesomeness of modern computer representations -- it sure beats using pi=3.14 or e=2.72 like I did in junior high; or looking up sines, cosines, and logarithms in huge, handwritten tables that often had copyist errors or mistakes in calculation in the less-significant positions, as mathematicians and engineers had to do for years/decades/centuries before the advent of modern computers.

When literally three of your four sentences are provably and factually wrong, why do you bother posting it at all?

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edit: miscounted your sentences, so fixed the numbering in my reply. Also added in critique of the second sentence, marked above.