The historical Jesus may well have made a claim to be the son of God, but what it meant to be a "son of God" is not set in stone and meant different things to different Christian writers. Being a son of God often came with the title of Moshiach/Messiah/Anointed One, and it was bestowed on many Israelite kings.

For example, Psalm 2:7 calls David a son of God, and this same honor is bestowed on Jesus by the author of Luke/Acts (Acts 13:33) and the author of Hebrews (Heb 5:5). Solomon was also a recipient of the honor (2 Sam 7:13-14).

Late 1st-century orthodox Christians saw Jesus as a special chosen one of God but there were certainly a wide variety of views as to what that meant. Paul, the earliest known Christian writer, doesn't call him a unique son of God but does call all believers sons of God. It is only 25-45 years after Paul's writings (Luke, Acts, John, 1 John) where we start seeing Christians who view Jesus as a unique son of God, even higher than David or Solomon but not necessarily the way today's mainstream Christians see him.

The views of what it means to be a son of God among the earliest Christians are different than the views of Christians that followed a few decades after, and these are just the differences we find in the canonical writings. Christianity was very diverse even before Paul came onto the scene (he even writes about people preaching other gospels before him and gets into doctrinal tussles with Peter), and they all didn't agree on who Jesus was or what he did in the primordial days of Christianity.

In summary, if Jesus saw himself as a messiah like David and Solomon, he probably also saw himself as--and called himself--the son of God. What exactly a "son of God" was varies among our earliest Christian writings.

I feel like I should say that whichever scholar you ask about this usually gives an answer according to their particular ideology. Someone like Bart Ehrman or Crossan would say Jesus didn't claim to be God because that's not what pious Jews did in the first century. Many people in historical Jesus studies will agree with them. Someone like N.T. Wright or Craig Evans would say Jesus did claim to be God because the early testimonies about him said so. It really depends who you ask. In that sense, it's a matter of debate or personal disagreement among historians. Even some new testament scholars will say that the resurrection can be historically proven based on the records.
Many of the new testament writings seem to give Jesus divine or semi-divine status. The gospels hint that he is God, only being explicit in the gospel of John. We can't verify whether these accounts are authentic because we don't have outside testimony about Jesus. It's really a matter of personal belief.

Nobody really knows what Jesus said or did not say. We only know what people in later generations claimed he said. Most critical scholars think it is very unlikely that Jesus claimed to be God. There's a lot of reasons for this. Bart Ehrman's How Jesus Became God is a good exposition on that subject. One of the reasons is that it would have been very un-Jewish, but a more significant reason is that the Synoptic Gospels never have him make that claim and always speak of Jesus and God as separate entities with separate wills.

The phrase "Son of God" is a phrase that's used in the Old Testament to refer to kings of Israel (Psalm 89, 2 Samuel 7:13-16, Isaiah 9:6) or other people specially chosen by God (Psalm 82:1-8). It's also used to refer to the Israelites collectively (Ex, 4:22), Ephrapha (Jeremiah: 31), any "righteous man" (Ecclesiastes 4:10), or whatever Genesis 6:2 is talking about. In the New Testament, the Gospel of Luke calls Adam "son of God" (Lk. 3:38). The Dead Sea Scrolls use it to refer to the Davic Messiah (4Q174, 4Q246). If Jesus claimed to be the Davidic Messiah, it's possible that he claimed to be the "son of God," but in his cultural and linguistic context that would have been a claim to be a human king, not a literal demigod status.

That article has a strange way of looking at it which assumes, I guess, that we know exactly what Jesus did and said (perhaps because they assume the Bible is revealed truth?) In actuality the Bible is a difficult and problematic source; a third century compilation of often-conflicting narratives about Jesus written decades after his death, which reflect the many different theological understandings and political goals within the rapidly evolving early Christian community. Reconstructing what the "historical Jesus" actually did and said from that is not impossible but it certainly isn't black and white.

This isn't my field and as others have said it's hard to find anything that resembles a consensus in historical Jesus studies, so I'll try to steer clear of giving you a single-source answer. However I will recommend Bart Ehrman's How Jesus Became God, which is an excellent read and addresses exactly this question of how Jesus came to be understood as God. Specifically, chapters 1–2 discuss what it meant to be considered "divine" in the classical Greco-Roman and Jewish worlds (a much more nuanced and flexible concept than today!); chapter 3 asks "Did Jesus Think He Was God?" (Ehrman says probably not); chapters 4–5 posits that the story of Jesus' resurrection (wherever it came from), planted the seed of the idea that Jesus was something other-than-human; and chapters 6–9 trace the development of this seed into the Nicene understanding that Jesus is the fully human, fully divine, co-equal and co-eternal single God that exists in three consubstantial persons (via a bewildering number of other considered-rejected-and-declared-heretical Christologies). It's a bit of a slog for a popular history book, and like I said I can't vouch for how well it represents current thinking in wider historical Jesus studies, but I'd highly recommend it if you're interested in this topic.

"Son of God" (uppercase "S"), maybe. Something like that would fit the blasphemy charges by the Sanhedrin. "son of God" (lowercase 'S"), almost certainly. Part of the message almost universally agreed to go back to the historical Jesus is the intimate familial relationship between the average Joe and God.