What's the real name of Safina?
Safina had before this incident a different name (Here an extract from a-Dhahabi's siyar a'lam an-Nubala'a):
وَسَفِينَةٌ لَقَبٌ لَهُ ، وَاسْمُهُ مِهْرَانُ ، وَقِيلَ : رُومَانُ ، وَقِيلَ : قَيْسٌ .
He was called Safina (as a nickname): and his name was Mihran, some said: Roman and some say Qays.
According to Arabic wikipedia the historian al-Waqdi was among those whom said his name was Mihran ibn Farrukh مهران بن فروخ .
قال الواقدي اسمه مهران بن فروخ،
While ibn Sa'ad said: his name was Roman and it was said Rabah and it was also said Qays and this is the more probable.
وذكر ابن سعد ان اسمه رومان ويقال رباح ، ويقال قيس وهو الأرجح .
While ibn Kathir in his al-Bidaya wa an-Nihaya called him Safina ibn Mafinah سفينة بن مافنة.
وذكر ابن كثير ان اسمه سفينة بن مافنة
I've also found other names but without a quotation of the source.
Sources of this narration
In the same book (al-Bidaya wa an-Nihaya البداية والنهاية volume 8) ibn Kathir -where he said that Safina is of a Persian origin, so it is unlikely that he had been black- quoted the hadith about his naming -without quoting a narrator chain- saying that at-Tabarani has also compiled or narrated it too (beside Ahmad who has compilled it in different versions and al-Hakim).
وروى الطبراني: أن سفينة سئل عن اسمه لِمَ سُمِّي سفينة؟
قال: سماني رسول الله ﷺ سفينة، خرج مرة ومعه أصحابه فثقل عليهم متاعهم، فقال لي رسول الله ﷺ: «ابسط كساءك».
فبسطته فجعل فيه متاعهم، ثم قال لي: «احمل ما أنت إلا سفينة».
قال: فلو حملت يومئذ وقر بعير أو بعيرين أو خمسة أو ستة ما ثقل علي.
I found it here in al-Mo'jam al-Kabir of at-Tabarani.
You may also find the version from abu Nu'aym's Hilyat al-Awliya' حلية الأولياء وطبقات الأصفياء here.
Al-Haythami quoted it in his majma' az-Zawaaid مجمع الزاوئد ومنبع الفوائد (this can be considered a hadith collection, but it is in first place a collection from original sources without quoting the full chain, but with a short comment like: where you may find it, a short qualification of narrators and sanity) adding the information that it was also compiled by al-Bazzar أبو بكر البزار (I his Musnad, see here) and that the chains of at-Tabarani and Ahmad have trustworthy narrators.
Comment about the provided link and claims
As already explained by @III-AK-III Safina was the person whom can be considered the source of this hadith and none of the versions seem to show that he disliked being named or called a"ship (=Safina)". The narration itself rather shows that he felt proud of being named this way, there are indeed other narrations where he refused to reveal his "real" name (before this incident).
Also note that the author of the linked article refers to Zaad al-Ma'ad زاد المعاد of ibn Qayyim al-Jawziya but is confounding between people whom have been slaves of the prophet Muhammad (or his wives) and those sahaba whom served him -willingly- like Anas etc. It also assigns slaves (like Bilal) whom have never been in the possession of our prophet () -falsely- to him. The hadith however is not quoted fully there rather than a short quote about Safina here (page 112-113) -Note the translation is mine-:
ومنهم أنجشة الحادي ، وسفينة بن فروخ واسمه مهران ، وسماه رسول الله صلى الله عليه وسلم سفينة ؛
Among them (refers to the mawali-freed slaves) Anjasha al-Hady and Safina ibn Farrukh and his name was Mehran. WHom the Messenger of Allah peace and Allah's blessings be upon him gave the name Safina (ship).
لأنهم كانوا يحملونه في السفر متاعهم ، فقال : ( أنت سفينة ) .
Because they used to let him carry their belongings during their travel. So he said to him: "You are Safina".
قال أبو حاتم : أعتقه رسول الله صلى الله عليه وسلم ، وقال غيره : أعتقته أم سلمة .
Abu Hatim said: He has been freed by the Messenger of Allah peace and Allah's blessings be upon him, while others said: he was freed by Umm Salamah.
So he was there said to be one of two whom used to carry the belongings of the prophet and those whom were in his company. He also forgot to quote that Safina was released -according ibn Qayyim al-Jawziyya- by the prophet himself or by his wife Umm Salamah, which would leave a totally different impression of Muhammad () for the non-Muslim readers. So basically one could say that this article is based on quotes taken out of context and used to show off that the Messenger of Allah () was a racist and the religion he was preaching for is racist.
In that article you may find this quote:
Muhammad had many male and female slaves. He used to buy and sell them, but he purchased more slaves than he sold, especially after God empowered him by His message, as well as after his immigration from Mecca. He once sold one black slave for two. His name was Jacob al-Mudbir. (attributed to Zaad al-Maad page 160).
From this I only found the first part, which was hardly conform to the Arabic original -note the difference-:
واتخذ الرقيق من الإماء والعبيد ، وكان مواليه وعتقاؤه من العبيد أكثر من الإماء (source)
- My translation take it carefully-
Muhammad had male and female slaves, and his freed slaves among male slaves were more than among the female slaves.
Ibn al-Qayyim then moves to explain that it seems as if the prophet () used to free for each female slave, two male slaves. He also tried to justify this by referring to this hadith from jami' at-Tirmdihi:
Any Muslim man who frees a Muslim man, then it is his salvation from the Fire - each of his limbs suffices for a limb of himself. And any Muslim man that frees two Muslim women, then are his salvation from the Fire ...
The rest of the quote of this author is imagination and a fairy tale which he had somehow brought together by mixing half facts with fantasy to create a -wrong- big picture.