I am currently reading David Vallado's 'Fundamental of Astrodynamics and Applications' and I have this doubt on the 2nd chapter named 'Kepler's Equation and Kepler's Problems'. Although I understand what true anomaly and mean anomaly (also, here is a good explanation) mean, I don't clearly understand the purpose of eccentric, parabolic and hyperbolic anomalies. Are they only for relating true and mean anomaly in different cases of eccentricity?
Yes, they're all just forms of the same relation that are specific to different values of eccentricity.
This is one of the few topics treated in greater detail in Richard Battin's An Introduction to the Mathematics and Methods of Astrodynamics (chapter 4, pages 141–173) than in Vallado, including more biographical anecdotes about the mathematicians who contributed to the problem. Battin's also a source (the second half of chapter 4, pages 174–190) of more details about the derivation and properties of Stumpff's use of Sundmann's transformation to create the "universal variables" formulation which unites the three sets of historically separate development, though Sconzo might be even better.