Putin refers to (if you are interested you can look up his statements and views on this, including video footage, that I linked below) recent research from Russian historians pointing to the "fact" that this story was fabricated by the Vatican emissary of Moscow at that time to hurt Russia's national interests, namely citing Vatican's interest in converting Orthodox Russia into Catholicism.
This is only one of the times Putin spoke about it, bit according to him the debate is live among historians. https://youtu.be/F0g07j4HohU
I spoke briefly about Ivan the Terrible here, but you're correct that the historicity of Ivan's murder of his son is in question--but the specific narrative here, that Ivan was painted as a violent leader by Catholics, is rather curious. But the general story that Ivan killed his son was very widely accepted until fairly recently.
If we turn, for example, to the Imperial historian Karamzin:
The Tsar struck him several times...and hit him hard in the head
Karamzin then turns to a rather fanciful characterization of Ivan begging for forgiveness, and Ivan Ivanovich accepting his death gracefully. Karamzin was an acolyte of autocracy, most especially that of Peter the Great, but regarded the collapse of the Rurikid dynasty rather poorly--so while he is the preeminent eighteenth-century historian, he isn't exceptionally reliable, and it is this version that crystallized in the nineteenth century. Certainly most Soviet historians made little effort to proffer a redemptive version of Tsardom, but more recently as you see there are greater efforts to reclaim this history.
Looking at the sixteenth century sources themselves, a murkier picture emerges: many chronicles reference only the death of Ivan, but make no intimation that this was at the hands of his father. Foreigners do allude to this, but offer no definitive or reliable claims--moreso reporting rumors. The Jesuit Antonio Possevino offers yet another story where Ivan had instead attacked his son's wife, precipitating the argument. This is the story accepted by the philosopher Mikhail Shcherbatov.
However: the dates on all these stories are contradictory, and there is a letter from Ivan the Terrible referencing his son's illness. The archival historian Nikolai Likhachev (one of the Imperial academics charged in the 1931 Academic Case purge) concluded that this was the most likely cause of death.
So, certainly some of Ivan's contemporaries, including but not exclusively foreigners, believed him to be responsible for his Ivan Ivanovich's death. Ivan was indeed a violent man, even if this was not uncommon for his time. But the veracity of this particular narrative is open to question.