I had thought that the Serbs had agreed on all but one, to let the AH police investigate in Serbia. However, My teacher insists that this is false and that the Serbs actually agreed with this, but disagreed with letting the criminals be tried in an Austro-Hungarian court instead of a Serbian court.
Is there eny truth to this?
To my knowledge (and I'm always ready to be shown otherwise if I'm wrong), you are correct. The sticking point was Point 6 of the ultimatum, specifically letting the Austro-Hungarians conduct an investigation inside Serbia. The Serbian government considered this a violation of its sovereignty, and given the Empire's view that Serbia should be crushed, the Serbians saw it as a slippery slope. First, the police come in...what next? The army? Governmental control? Absorption with semi-autonomy a la Hungary? All of which were non-starters for Serbia.
On the other hand, Austria-Hungary already believed that the Serbian government was complicit, or at least had turned a blind eye, to the assassination of the Archduke. If that's the position that the Empire starts from, then letting Serbia conduct its own investigation is similarly a non-starter, as the Empire believes the results of the investigation will be skewed, if not outright lies.
There is no doubt that some (perhaps many) in the Empire's governmental structure, especially Conrad von Hotzendorf, wanted to craft the ultimatum in such a way as to provide an excuse for war, with Austria-Hungary appearing the aggrieved party. And with Franz Ferdinand dead, the best voice among the Empire's powerbrokers for relative pacifism and for Serb autonomy/independence was eliminated, meaning that Hotzendorf and his ilk were likely to get their way.
Nevertheless, in my view, I don't think that is mutually exclusive with Austria-Hungary's position that any investigation conducted solely by Serbia would be flawed from the start. I also think that Serbia's agreement to the ultimatum on all points except Point 6 only confirmed the Empire's thought process. Further, the Serbian response was that they would, in some cases, share the results of their investigation with the Empire, seemed to fuel the thought that some aspects of the investigation would be hidden.
Not only that, but many of the conspirators were actually tried in Sarajevo, which was in the Empire. Some did of course escape and were ultimately tried on the Salonika front in 1916-1917, but I think that's a separate issue.
Sources: "Catastrophe 1914: Europe Goes To War" - Max Hastings "The Sleepwalkers: How Europe Went to War in 1914" - Christopher Clark "The Guns of August" - Barbara Tuchman "The War that Ended Peace" - Margaret MacMillan