I'm not an expert, but I just read a book about the history of the Teutonic Order a few months ago, the parts I found relating to your question are:

The most important description of the battle is that of Jan Długosz, the Polish court historian. It is brief and tends to glorify the Polish contribution to the victory at the expense of the Lithuanian. In sum, he wrote that one wing of the 'crossbearers' defeated the horsemen under Vytautas after fierce fighting. Although Vytautas and the Smolensk regiments remained on the field, the Tatars fled, followed by many Lithuanians and Rus'ians. The German crusaders, seeing the wild flight of the enemy, assumed they had won a victory and left their positions to pursue them. This left a gap in the order's lines. The Poles meanwhile, had been holding their own against the Teutonic Knights. Now seeing their opportunity, they pressed harder, and came in through the gap created on the left when the crusaders broke ranks to pursue the Tatars; soon the Polish knights had put the main battle force of the Teutonic Knights in great difficulty.

This generally accepted understanding of the battle has been modified significantly by a recently discovered letter written in 1413 by a well-informed noble or mercenary captain. Its finger-wagging admonition to keep the ranks of the knights firmly in hand supports an alternate version of the combat given by less well-known chroniclers, that a small number of crusaders attached to the Teutonic Knights had fallen for a tactical ruse by the Lithuanians, a feigned retreat that led pursuers into a trap sprung by Polish knights waiting on the flank. The Lithuanians and Poles then drove into the disordered lines and rolled up the crusader formation.

A bit from the next paragraph, describing some effects of Ulrich von Jungingen (commander of the crusading force) charging into the Polish lines hoping to get to Jagiello:

Vytautas, seen at the center of Matjeko's painting, had seemingly been everywhere at once on his wing of the battlefield, performing fantastic and courageous feats; he now hurried to over to the royal position with his men, perhaps to urge the king to reinforce the main battle lines with his reserves.

Modern scholars, despite new archaeological information and newly discovered archival material, have not come to complete agreement as to what transpired. ...Because everyone agrees that the Teutonic Knights' defeat resulted from the ill-disciplined pursuit of the Lithuanian forces, the dispute about the motivation of the Lithuanian units cannot be resolved to universal satisfaction: either there was a strategic retreat on the part of a significant fraction of Vytautas' forces or those Lithuanians, Rus'ians, and Tatars had been driven from the field in defeat.

The author also says that regardless of whether or not it the Lithuanian retreat/ruse was purposefully done, the resulting Teutonic pursuit/break of formation contributed to their defeat. Unfortunately I couldn't find the "noble or mercenary-captain"s account in the list of sources. Hopefully this helps to answering your question and sorry if it isn't up to this subreddit's standards. If you'd like there are a few more paragraphs I could type out tomorrow dealing with more of the battle.

The Teutonic Knights: A Military History by William Urban

The Annals of Jan Długosz English abridgement by Maurice Michael.

Battle of Grunwald by Jan Matejko. Vytautas is the fellow in red at the center: https://upload.wikimedia.org/wikipedia/commons/a/a1/Matejko_Battle_of_Grunwald.jpg