But my question isn't about the story of Gajendra, but rather about the
one who saved him. Strictly speaking, it was not Vishnu himself who
saved Gajendra, but rather Vishnu's incarnation Hari, as described in
the Srimad Bhagavatam:
First of all there is no difference between Lord Vishnu and his incarnations.
We know that Hari rode on Vishnu's bird Garuda, although that's not so
unusual since Krishna also rode Garuda, as I discuss in this question.
(Rama didn't ride Garuda, but he did meet him; see my answer here.)
As per Valmiki Ramayan, when King Dasharath arranged Yagya, all deities headed by Brahma praises Lord Vishnu and did complaint regarding sins committed Ravan by Ravan who can only be killed by human being as per boon granted by Lord Brahma. Lord Vishnu promised them that he is gonna born as son of King Dashrath and will kill Ravan.
So, Lord Vishnu appeared as Shri Raam in TretaYug. Similarly as per VedVyass Mahabharat, Amshavataran Parva, In DwaparYug, Lord Vishnu promised all deities that he gonna born as Krishna, the son of Vasudev and Devaki.
So, does Raam and Krishna are different from Lord Vishnu? Nope! Same personality played role of Raam, Krishna, Narasimha etc etc based on timing and circumstances. So it is not fair to differentiate among various incarnations assumed by Lord Vishnu.
If you wear white colored shirt on Monday and Red colored shirt on Tuesday, that does not mean that @Keshav Srinivasan on monday is different from @Keshav Srinivasan on tuesday. Personality is same, but forms are different. Eg. Discussion between Shri Raam, Shri Lakshman and Parshuram is leela only by which human beings can learn something from that conversation.
But don't bind the Supreme Lord by rules of physical body of human being. Lord of Universe can do whatever he want. Whatever he does, everything has special purpose. He has ability by expanding himself in multiple forms to perform various tasks.
In Raam Avtaar ( when he descended as Raam ), as being Maryada-purushottam, he never used his extra-ordinary powers in whole his life but he started to show his extra-ordinary powers since childhood when he descended on earth as Krishna.
He is antaryamin, still why couldn't he identity who is Vaali and who is Sugriv during their battle? Because he had to give lesson to human beings of wearing garlands in such situations. For reaching to Shri Lanka, he never used Garud but gave opportunity to his devotees to build Raam-Setu. When he descended as Kapila, Matsya, Kurma, Varah, Vaman, Parshuram etc. Did he used Garuda? His every action is to guide human being as he confirmed in BhagwadGita.
Hence, you already know story of GajendraMoksha, and now you also know that Hari and Vishnu are not different. Hence, you got answer to your question.
Mahabharat UdyogParvai.Yanasandhiparva, 77 mentions meanings of various names of Lord Krishna such as Narayan, Hari, Madhusudan, Janardan, Madhav, Keshav etc.
When Vibhishana arrived to join the army of Lord Shri Raam that time Sugriva doubted that Vibhishana might cheat them. Hence he expressed his views in front of Lord Shri Raam and Supreme Lord Shri Raam replied.
तत: शुभतरं वाक्यमुवाच हरिपुङ्गवम् |
सुदुष्टो वाप्यदुष्टो वा किमेश रजनीचर: || 22 ||
सूक्ष्ममप्यहितं कर्तुं ममाशक्त: कथंचन |
पिशाचान्दानवान्यक्षान्पृथिव्यां चैव राक्षसान् || 23 ||
अङ्गुल्यग्रेण तान्हन्यामिच्छन्हरिगणेश्वर |
श्रूयते हि कपोतेन शत्रु: शरणमागत: || 24 ||
Here Lord Shri Raam declared directly that he can kill all of the pishacchas, danavas, yakshas, rakhasas present on the earth by tip of his finger.
Hence, through incarnation known as Shri Raam, Lord Narayan ( Vishnu ) had played leela like simple human being without using his extra ordinary powers.
In 4th Charan of both TretaYug and DwaparYug, Deity of earth with other Devatas headed by Lord Brahma approached to Lord Narayan and glorified him by Purush Sukta of Vedas and requested him to save themselves from Kings of Demonic nature ruling over earth and other lokas.
That time Lord Vishnu promised them that he will incarnate as son of Dasharath and Kausalya ( In TretaYug ). And Same Lord Narayan promised them that he will incarnate as son of Vasudev and Devaki ( in DwaparYug ).
So, Lord Narayan played the role of Shri Raam in TretaYug. Same Lord Narayan played role of Lord Krishna in DwaparYug.
In TretaYug, he didn’t used his extra-ordinary powers at all because he want to give ideal example of “Maryada Purushottam” for human beings. Similarly by hard penance, Ravana asked for boon that he can be killed by Vanar or human being only, not even Devatas can kill him. Prajapati granted him such boon. Hence, Lord Narayan incarnated as human being as per request of Devatas who were getting troubled by Ravan. Those were the reasons why Lord Naraya was not using his extra-ordinary powers when he descended as son of Dasharath and Kausalya.
Since Lord Narayan is antaryamin of each of the Jiva, so he could easily identify who is Vaali and who is Sugriv? He could do so, but what human being will do in such situation? Hence Lord used idea of garland ( Maala ) so that ordinary human being will learn what to do in such situation.
Same Vaali born as Vyaadh ( hunter ) in DwaparYug who shot the arrow on toe of Lord Krishna accidentally. At that time hunter started to apologize..Lord Krishna revealed his 4 armed Narayan form and told, ”Don’t worry. In TretaYug, I shot you with arrow from behind. I wanted to give you justice, hence in this DwaparYug, I let you shoot me by arrow. No need to apologize. Within few momments, Vimana will arrive and will carry you swarga Loka alive.” By saying so, Lord Krishna went back to Vaikuntha ( Avtaar Samapti ).
Even Lord Raam could fly up and bring back Sita from Lanka but he didn’t do so. Because he wanted to teach ordinary human beings what to do in such situations.
But when Shri Raam came as Shri Krishna, he started to show his powers since childhood, such as killing Putana, Kaliya, Dhenukasur, arishthasur, lifting up Govardhan mountain etc etc.
Shri Raam is Maryada Purushottam avtaar.
Shri Krishna is Purna Purushottam avtaar.
Both roles are played by Same Supreme Being. People who deeply studied Ramayna, Mahabharat, Vishnu Puran and Srimad Bhagwat can understand it properly.