Forgive me if this doesn't necessarily answer your stated question, but I believe it solves the real question of the issue here.
There is no debate going on in the crowd. John is simply quoting what the people were saying. But what the people were saying were in no way competing identities. Rather, they were different titles for the same figure, but with possibly slightly different expectations of what that meant.
First, I need to remind the reader that Christ (Christos) is the Greek title for the Hebrew Messiah (Mashiach). They both mean "Anointed One".
So for a second Moses, The Prophet, to be different from the Christ would mean they were saying the second Moses was different from the Messiah. As we will see later, this is not the case, they are the same, but keep this in mind as we continue. I will be often simply using Messiah from this point on rather than Christ.
If we read further in the chapter, the Pharisees themselves make this evident.
40When they heard these words, some of the people said, “This really is the Prophet.”
41Others said, “This is the Christ.” But some said, “Is the Christ to come from Galilee?
They replied, “Are you from Galilee too? Search and see that no prophet arises from Galilee.”
Here we see that the Pharisees conflate the two into one identity.
That is, one group calls him prophet and the other Christ[Messiah] but comment the Christ[Messiah] doesn't come from Galilee. Then the Pharisees say no Prophet comes from Galilee.
The NIV margin note indicates that two early manuscripts (p66 and p75) read "the Prophet" instead of a prophet. A reference here to "the" prophet fits the context well (v. 40) and has been accepted by a number of scholars. 1
Besides telling us that the crowd is clearly well educated on these matters (as they spot it as quickly as the Pharisees), this tells us the two titles of Prophet and Christ[Messiah] were referring to the same person.
Why different titles?
While the people were maybe not disagreeing on the basic identity they were connecting with Jesus, the different titles they used did perhaps communicate different expectations or connotations of what they meant.
Jewish eschatology was no less confusing or debated than Christianity's today. In fact, one could argue it was worse.
This article lays out, from Jewish sources, all the different roles that were expected from the Messiah. Messianic Titles.
The page is actually all the source material for an article, linked on that page, that speaks to how Jesus fulfilled or will fulfill the roles.
The classic phrase "Prophet, Priest and King" captures most of them. But I will give a key quote you may find in that article:
"It would be a fatal error to assume that investigation of later Jewish exegesis of Dt. 18:15, 18 settles definitively the question whether the figure of Moses influenced Messianic expectation. In many passages the Messiah is depicted as a second Moses even though there is no reference to Dt. 18. These are passages which have as their basis the doctrine that the redemption out of Egypt is a type of Messianic redemption. Taught by the OT itself, this doctrine “as no other” “comprehensively determined at an early period the shape of the teaching concerning the final redemption.”" 2
So while the title prophet or The Prophet may refer to Moses, but it may be simply identifying Jesus as a prophet. Regardless, the Messiah was often seen as a prophet, one way or the other.
The kingly role of the Messiah may have been being referenced to the exclusion of the Prophet. This is, in fact, the most basic meaning of "Messiah" as Anointed Davidic King.
Those who called Jesus the "Christ" may or may not have included "Prophet" in his role as Messiah.
The crowd was not debating whether Jesus was saying he was the Messiah. No, they are united in that and the Pharisees also clearly see this.
However, the different titles simply communicate a tiny bit of the eschatological debate of the time in how they viewed that Messiah.
Unfortunately, claiming what those expectations were exactly would be speculation. I again recommend the ChristianThinkTank article to read about all the different possibilities.
1) IVP New Testament Commentaries
2) [Kittel, G., Bromiley, G. W., & Friedrich, G. (Eds.). (1964–). Theological dictionary of the New Testament (electronic ed., Vol. 4, pp. 859–860). Grand Rapids, MI: Eerdmans.]