The Greek word translated as "glorify" and "glory" are from the same Greek root, just like our words "glorify" and "glory" are from the same English root. There translated as "glory," however, is misleading. The English words were chosen more to praise Jesus than to translate his words. At best, the "glory" aspect of these words in tangential.
A more accurate translated of these words in today's English would be "recognize" and "recognition." These words provide cover both the primary and secondary meaning of these Greek word s Jesus used them,
The Greek word translated as "glorify" is doxazo, which primarily means holding something or someone in your mind or imagination. Its secondary meaning is to "extol" or "magnify," which is where the meaning of "glorify" as in "praise" arises. In the Latin Vulgate the Greek word was translated as clarifico, which means "to make illustrious or famous." Interesting, this Latin word is also the source of the English word, "to clarify," which goes back to the primary meaning of the word, having a clear image in your mind.
The noun form is "glory" in English. This is from doxa, which means "expectation" and "opinion." It came to mean "reputation," especially "good repute", "honor", "glory" and rarely "ill repute." It came to mean "glory" and "magnificence" in external appearance through Christian writing after the Gospels were written. Christ did not seem to mean it that way, at least not exactly.
Generally, in the alternative translations of Christ's words on this site, the most common word used to translate doxazo is "to recognize." This idea is also works for the noun form as "recognition." This idea of "recognition" captures the mental imaging part of the concept and the idea of giving honor. By giving people recognition, we make them better known and more broadly respected. People who are more broadly recognized are more famous.
To maintain the connection between the noun and the verb, the noun should be translated as "recognition." While "reputation" is more accurate, "recognition" also has the sense of praise in English.
Today's processes of promotion and advertising are the most common forms of "glorification." Advertising is used to create images in our mind, making people and tings more illustrious and famous.
However, in Jesus's era, the process was almost entirely word-of-mouth, much like the communication processes we see on the Internet today: people sharing information with others because it is interesting. The process of making someone or something more broadly known is what Christ describes by doxazo, that is, glorification.
We could use "advertise" or "promote" in the place of "glorify" in today's translations of Christ's word, but many would certainly think of that as disrespectful.