Hey, you never got an answer it seems but I randomly happened upon this and I've written my thesis about Sol and its current proposed aspects (Sol Indiges, Sol Invictus Elagabal and Sol Invictus) so I feel I can offer some answer here!

First off, I'd add that like many parts of history and in historiography there's going to be different views and ideas on how things were, this also includes the god Sol.

One main thing that's changed from 20th century scholarship to now is the idea that there was a roman god, Sol, which we now identify as Sol Indiges to distinguish him from that of Sol Invictus and Sol Invictus Elagabal. Sol Indiges was a native god of rome, and while nowhere as dominant as Jupiter, Mars, etc, had a fairly stable following throughout the republic and into the early age of the empire. We know Augustus for instance dedicated two obelix' captured from the conquest of Egypt in Sol's name. He was the patron god of the Circus Maximus and the protector of chariot riders.

Sol Invictus Elagabal, however, is with pretty certain assurity a syrian god based on the Emese (modern day Homs) god of Elagabal (Elah Gabal, the king of the mountain). He came into prominence in Rome by the ascension of emperor Marcus Aurelius Antonius, nicknamed Elagabalus/Heliogabalus (I will call him Heliogabalus in order to differentiate him from the god Elagabal).

Heliogabalus was a 14 year old priest-king (dedicated to Elagabal) of Emese who was, through the machinations of his grandmother Julia Domna (the wife of the late emperor Septimus Severus), claimed to be an illegitimate son of the recently murdered emperor Caracalla -- who he actually was a distant cousin of, not bastard son.

Upon his crowning in Rome he took with him the worship of Elagabal, a black rock and other cultic objects. He made a new temple on Palatine hill in Elagabal's honour and titled himself sacerdos amplissimus dei invicti Solis Elagabli, "the elevated priest of the unconquered sun god Elagabal"

Heliogabalus reigned in 3 years and most history we have (written afterwards) paint him in a rather pathetic and cruel light) -- but from what he know he offended the elites of roman high society by refusing to wear traditional, roman clothing and his decree to make Sol Invictus Elagabal the supreme god of the pantheon angered quite many. He even gave his grand-mother and and mother a place in the senate which angered even more.

After 3 years he was murdered by his own praetorian guard and the senate quickly decided to issue a damnatio memoriae on him and Sol Invictus Elagabal - which meant that his name, image and legacy were to be stricken away from everything. Sculptures bearing his image would be destroyed, objects with engraving or carvings of the sun god or the late emperor would either have the text removed or destroyed and so on.

It is safe to say that the cult of Sol Invictus Elagabal did not flourish after that time and we know from archeology that no surviving dedications or much praise was given to Elagabal in rome after this time. In syria the god continued as before, and the black stone Heliogabalus took to Rome was returned to its temple there.

The case of Aurelian is different, however. Some historians believe his reign reintroduced the syrian god Elagabal, but recent findings and more historians assess that Aurelian's reforms regarding Sol Invictus was not the same as the god of Sol Invictus Elagabal. Some, like Steven Hijmans, would say that this is actually the roman god Sol Indiges that's being elevated by Aurelian.

But it's important to note the Aurelian did not make Sol Invictus the "king of the gods", what he did instead was to make all gods set on equal terms in regards to the panteon - but then made Sol Invictus the official god. He even sidestepped the entire traditional college of pontifex by creating and entirely new one dedicated to Sol Invictus, the Pontifices Dei Solis.

Aurelian's motivation in doing this was probably not of zealotry like that of Heliogabalus, but rather a political maneuver in order to strengthen his position and the authority of the Emperor and state, which had seen a severe dip in credibility and position during the era of the Soldier-emperor's and the chaos that had followed. He wanted to unify and strengthen the state and by making Sol Invictus its sponsor and source of power and moral bond -- much mimicking the efforts done by Augustus, hundreds of years ago.

Which is why it is very likely that Sol Invictus of Aurelian times was not the same, syrian-based god of Sol Invictus Elagabal of Heliogabalus. Aurelian was a staunch roman traditionalist and mirrored much of his work on Augustus. It wouldn't make much sense that he had taken a syrian god as his sponsor then.

So in conclusion: Heliogabalus was the first to try to change and usurp Jupiter of his position as head of god, by introducing the syrian Sol Invictus Elagabal at the throne. This, however, only lasted three years and was quickly reverted after his death. Fifty years later, Aurelian came to power and while he did not make Sol Invictus the head of the panteon - he made him the official god of the roman state (and emperor), while making the rest of the gods to be on equal footing. Some argue this as a prelude to creating monotheism in the roman empire, but that's heavily debatable if it ever was. Anyhow, Sol Invictus was never the "head of the gods" after Aurelian but rather their official gods which the following emperor's took into varying degrees of enthusiasm, Constantine the Great being among its greatest champion until he turned to christianity.