I think you have the direction of its evolution reversed.
Both dedicate and inaugurate originate in Latin and have similar meanings then and today
dedicate means to consecrate, proclaim, affirm, set apart.
inaugurate means to install or consecrate.
For dedication as in the dedication for a book, this a message at the beginning of a book saying that it was written or is being performed in order to honor or express affection for someone.
If you research the meaning of consecrate, you'll find this, where #2 may match the book dedication:
- to make or declare sacred; set apart or dedicate to the service of a deity: to consecrate a new church building.
- to make (something) an object of honor or veneration; hallow: a custom consecrated by time.
- to devote or dedicate to some purpose: a life consecrated to science.
- to admit or ordain to a sacred office, especially to the episcopate.
- to change (bread and wine) into the Eucharist.
Dedicating a park, bridge, or building fits all of this. If you dedicate something to someone, you may be doing to honor that person or you may devoting it to that person (for some purpose).
When you inaugurate something or someone, this may fit the definition of install (to establish in a place, office or position). Presidents are inaugurated. Rides at Disneyland are inaugurated. On the other hand, you may inaugurate a memorial site, in which case you may also be consecrating that site.