What you have read is right. The Valmiki Ramayana mentions the activity Hanuman did during his childhood.
When Jambavan narrates the birth story of Hanuman and familiarizes his power and the boons he had received from various celestials. Jambavan narrates how he got the name Hanuman too. From Kishkinda Kanda chapter 66
अभ्युत्थितम् ततः सूर्यम् बालो दृष्ट्वा महा वने |
फलंचेतिजिघृक्षुस्त्वमुत्प्लुत्याभ्युत्पतोदिवम् - यद्वा -
फलम् च इति जिघृक्षुः त्वम् उत्प्लुत्य अभिउत्पतो दिवम् || ४-६६-२१
Thereafter, when you were a boy you have see the just risen sun in the vast of forest, and deeming it to be a just ripen reddish fruit, and thinking it to be a best catch, you hopped up and flew towards sun on the sky...
शतानि त्रीणि गत्वा अथ योजनानाम् महाकपे |
तेजसा तस्य निर्धूतो न विषादम् ततो गतः || ४-६६-२२
Even on your flying up three hundred yojana-s, oh, brave monkey, and even when sun's radiation puffed and blown you down, you did not get into any anguish as were indefatigable.
त्वाम् अपि उपगतम् तूर्णम् अंतरीक्षम् महाकपे |
क्षिप्तम् इन्द्रेण ते वज्रम् कोप आविष्टेन तेजसा || ४-६६-२३
Oh, mighty monkey, even on seeing you who are coming near the vault of heaven, Indra is captivated by anger and by his effulgence he catapulted his Thunderbolt at you.
तदा शैलाग्र शिखरे वामो हनुर् अभज्यत |
ततो हि नाम धेयम् ते हनुमान् इति कीर्तितम् || ४-६६-२४
Then, your left cheekbone, zygomatic arch, is broken when you fell down on a highest mountaintop, from thereupon your name-phrase is indeed averred as Hanumaan.
This was again repeated in brief in Yuddhakanda chapter 28 when one of the aides of Ravana named Shuka describes Hanuman and his valor.
पतितस्य कपेर् अस्य हनुर् एका शिला तले |
किंचिद् भिन्ना दृढ हनोर् हनूमान् एष तेन वै || ६-२८-१५
"One of the jaws of this monkey who fell down on the head of the rock, was a little fractured. For this reason, because of his strong jaw, he is called Hanuma."
This(trying to eat Sun and faling down on a rock from a great height) was the activity Hanumaan was mentioning to Sita in Lanka while introducing himself in Lanka.