It is often very difficult or almost impossible to reconstruct what actually happened in the pre-modern battlefield due to either the scarcity of the trustworthy sources or their conflicting description. The memory of the battle itself is also able to have been fabricated in course of the time, sometimes with almost utterly different historical significance. The battle of the Ice certainly belongs to such a category of the battle in the past that had a very remarkable 'afterlife', or the later development of the tradition after the battle itself in 1242.

 

John Fennell, the classic non-Russian authority of medieval Russian history, even states that: 'Many of the description of the battle itself - in the Life [of Aleksandr] and in the later chronicle versions - are decorated with details which are clearly no more than fruits of the writers' imaginations' (Fennell 1983: 105). It should also been emphasized that there were not a small numbers of the chronicles written in medieval Russian towns, and they were not properly redacted for academic-use so out of Russian language that this complexity of the primary sources make it further difficult for non-specialists (including me who cannot read Old Russian by myself) to understand the exact situation on the battle itself as well as on the relevant sources.

 

Wikipedia's English article on the battle seems to be in fact based on the article by Donald Ostrowski (linked below), one of the few non Russian specialists in early Moscow state (Ostrowski 2006). In this paper, Ostrowski checks many available sources written both in near contemporary or in later period, and classifies the development of the historical tradition of the battle in the five phases in total. What I put down below is just a very crude excerpt from the discussions found in his paper:

• The oldest layer of the accounts, written within two generations after the battle (up to ca. 1300), does not explicitly state that the battle occurred on the lake, but rather by the lake. To give an example, Livonian Rhymed Chronicle (compiled in ca. 1300), German account of the Northern Crusade, describes that the dead fell on the grass (Ostrowski 2006: 291f.). Alternatively, one Russian sources mentions that Aleksandr chased the enemy across the lake, though another one even pay little attention to the participation of Alexsander, the hero of the battle in later traditions.
• The second layer of the accounts, dated from early 14th century, first specifies that the battle actually happened on the Ice, and some of the sources compare Aleksandr with Moses in OT's exodus. According to Ostrowski, this comparison also paved the way to the later addition of the dead soldiers falling through the Ice, as the army of the pharaoh suffered in Exocus 14:27-28.
• The description of the drown Germans can be first discerned in the 4th phase of the tradition, dated to the middle to late 16th century, best testified by this famous illustration found in the manuscript of the Life of Aleksandr compiled in ca. 1570. He continued to play an important role as a hero of Orthodox Christianity (against Western) also in Early Modern Russia and became further popular at that time.

 

Thus, as OP wondered above, the description of the drown soldiers through Ice was not entirely Eisenstein's invention (The 5th, last phase of the tradition according to Ostroski), bur neither it dates back to the early, trustworthy account of the battle.

  [edited]: adds an illustration depicted in 16th century.  

References:

• Fennell, John. The Crisis of Medieval Russia 1200-1304. London: Longman, 1983.
• Ostrowski, Donald. 'Alexander Nevskii's "Battle on the Ice": The Creation of A Legend'. Russian History/Histoire Russe, 33, Nos. 2-3-4 (Summer-Fall-Winter 2006), 289-312.