Can you give an example of the Mongol army under Genghis Khan breaking such a promise?

The two examples you gave for /u/unphilievable are mostly the same, though I don't know which book you read. The destruction of Samarkand and Baghdad were examples of the Mongols being the Mongols. They didn't retaliate against people who didn't follow through with promises, they retaliated against people who didn't follow the same rules of warfare as they did. In both of the examples below, the government of the opposing people group did not react well to diplomacy.

Samarkand

Samarkand was the a city of the Khwarazmian empire. If you don't know anything about this empire, you really should. They were an interesting and mostly short lived empire.

In 1218, Genghis Khan had wiped out or assimilated the Kara Khitai and sought to open up trade routes through the Khwarazmian empire. He sent an envoy of 100 muslim merchants to the city of Otrar. This is the best map of the empire that I could find. It doesn't show Otrar, but it does show other major cities such as Samarkand and Bukhara. It is superimposed over modern day country lines. The merchants were sent to Otrar, but were robbed on the road. They went to the leadership of the city of Otrar. The governor, Iralchiq said that the men should be arrested for spying. What proof he had, I'm not sure. He arrested the merchants and killed all of them.

This was truly an international event. One empire had killed the emissaries of another. Genghis Khan sent a message to the Shah, Mohammed II of Khwaraziem. It is believed that he received no answer from the Shah. It is a point of honor now that the Mongols must prepare for war. The governor of Otrar should pay for his crimes.

In 1219, the Mongols move into the territory of the Khwarezmian empire. He has approximately 100,000 warriors with him. The Shah has a much larger army, but they have no unit cohesion and are very poorly trained. The Mongol commanders are Ogodei and Chaghatai. They destroy Otrar, raze it, and continue to Bukhara. In 1220, the citadel falls and the city surrenders. The Mongols plunder the city, but they don't slaughter all of the inhabitants.

Samarkand is the next target. The population of Samarkand were mostly Persians. It's been explained to me that the Persian fighting style and thought process at this time was to fight until you cannot fight anymore, then seek peace. The Mongol fighting style was to offer peace first, then brutally slaughter your enemy. You can see where this might get the Shah in trouble. He is described as an unpredictable ruler, a coward, an untrained fighter, and even his own people preferred the Mongols to his treatment. That is really saying something if you are worse than the Mongols! On several occasions, the Shah's forces deserted to the Mongols. I don't have numbers for the city, but you can guess that it was heavily populated. By modern estimates, though, when the city was eventually repopulated, only a quarter of the city was inhabited.

In short, the city of Samarkand put up a fight, but the city was cut down. The Mongols looted, killed, and destroyed the city for 5 days. The only people spared were the artisans, who were shipped to Mongolia. Specifically in the case of Samarkand, the Mongols did not lie to the people of the city. They offered peace, which is the traditional Mongol way. When the peace was rejected, the Mongols killed almost everyone.

Baghdad

Baghdad is in the outlying portion of the Khwarezmian empire when Hulagu invaded. The people who inhabited this region remembered the time before Genghis Khan arrived with his Mongol hordes. They tried to re-establish a Muslim Caliphate in the area. In 1257, a full 40 years after the destruction of the Khwarezmian empire, Hulagu was sent by Mongke Khan to reclaim this territory for the Mongols.

The Caliph of Baghdad, Al Mustasim (this is probably spelled wrong), rejected the Mongols offer of peaceful submission and war ensued. The Caliph demanded that the subjects should retain their political and religious rights. Hulegu was not having that. The caliph is also described as arrogant. He claimed that the whole of the Muslim peoples would come to his aid if the Mongols attacked him. This was a giant bluff. There was not way he could have gotten enough messages out with the Mongols surrounding the city. Hulegu threatened that all of the humiliation that came to the former Khwarezmian empire would rain down on the Caliph ten fold. He also is reported to have said, "Open your gates and you will live." The Caliph did not open the gates and you can guess what happened.

In February 1258, the walls were breached and the people were slaughtered for 17 days. It took 17 days to round up all estimated 90,000 people and kill them systematically. The Caliph died as all 'royalty' do in the Mongol world, death by trampling. The city was sacked and burned.

Again, this isn't a situation where the people of the city surrendered then were slaughtered. The Caliph rejected the Mongols offer of peace and his city was sacked.

I'm not sure what situation you are referring to in your original question regarding the Song dynasty and help in the middle east. I'm sorry I can't answer that portion of your question.

Sources

Luc Kwanten's Imperial Nomads, 1979.

Michael Prawden's The Mongol Empire: It's Rise and Legacy, 1961.

Bertold Spuler's History of the Mongols, 1968.

Urgunge Onon's translation and commentary on The Secret History of the Mongols.