Is the Tokai Maru the ship that rescued many refugees from the 'Smyrna catastrophe' in 1922 or is it named Tokei Maru instead?

No. If there was this ship, then the name of the ship in question was Tokei Maru.

The reason for, and meaning of, "if" is illuminated in the first quote below: there it says that numerous reports make the whole story "plausible", although individual accounts have to be taken with a grain of salt, as they are inconsistent in some of their details to each other.

Although Tokai Maru was also a ship name used around that time.

As noted in the first comment below the question by @KillingTime: the ship called Tokai Maru mentioned on Wikipedia was launched only in 1930, making it quite impossible logically that it played any role in 1922's Smyrna events. The ship inquired about seems simply to not have a dedicated Wikipedia page. Yet.

An early Greek newspaper (shown in the National Herald Newspaper) renders the ship's name as Τόκει Μαροῦ:

— Astrapi-ΑΣΤΡΑΠΗ-26081922-ΑΦΙΞΗ-ΤΟΚΕΙ-ΜΑΡΟΥ.jpg

The Japanese researcher on the matter, Nanako Murata Sawayanagi — ("fluent in English, Japanese, and Greek") — said:

The Greek newspaper Empros reported the incident with the headline, “Massacre in Smyrna: Japanese Act Bravely,” saying that “The brave actions by the captain of the Tokei-maru are worthy of note. Despite being threatened by the Turks, he succeeded in rescuing 825 of our Greek comrades. After he unloaded all the small boats aboard, the Kemali soldiers surrounded them, threatening to sink them. In response, the captain told them, ‘If you dare touch a hair on the head of any of the refugees, I shall regard it as an affront to the Japanese flag and a threat to the Japanese government.’”[7]

An article entitled “Massacre in Smyrna: Japanese Act Bravely” was printed in the Greek newspaper Empros on September 4, 1922 (September 17 in the Gregorian calendar), describing the efforts by a Japanese freighter to rescue Greek refugees.

[…] The fact that reputable media and diplomatic sources mentioned it several times leads to its plausibility. Perhaps because it happened amidst the chaos of war, there are inconsistencies among the reports about the dates and the number of ships, making it impossible to take any one particular account of the rescue by the Japanese as a “historical fact.”

Footnote [7]: “Massacre in Smyrna,” Empros, September 4, 1922.

— Nanako Murata Sawayanagi: "Did a Japanese Ship Rescue Hundreds of Greek Refugees from Smyrna a Century Ago?", We Are Tomodachi, The Government of Japan, 2018.

This researcher also wrote earlier:

There is a “Tokei-Maru”[東慶丸]in the list of the PNS owned by Dairen Towa Kisen Company[大連東和汽船株式会社]. Its port of registry is Dairen (Da Lien City), now in China, but at that time in the Kanto District under Japanese control.²⁵ I have also found the ship “Tokai-Maru”[東海丸], which sounds close to “Tokei-Maru”, owned by Nansho Yoko [ 南 昌 洋 行 ]. Its port of registry is Susami, Wakayama in Japan.²⁶ The TNS also includes “Tokei-Maru” and “Tokai-Maru.”²⁷ I have collated the ship numbers[船舶番号]and signal letters[信号符号]of “Tokei-Maru” and “Tokai-Maru” in the list of PNS with those of “Tokei- Maru” and “Tokai-Maru” in the list of TNS, and concluded that “Tokai-Maru” in PNS and “Tokai-Maru” in TNS are not identical while “Tokei-Maru” in PNS and “Tokei-Maru” in TNS are the same. “Tokai-Maru” in TNS was owned by Tokai Kisen Company[東海汽船株式会社] and its port of registry is Kobe.²⁸

As the TNS shows the record of the ships that existed in 1923 though it was published in 1926, it can be said that “Tokei-Maru” existed at least between 1920 and 1923. That means that there is a possibility that this ship was at the quay of Smyrna during the catastrophic event in September 1922. The problem is that we do not know if one of its routes included Smyrna and if so, if it was there the day of the destruction of Smyrna. With regard to “Tokai-Maru” in TNS, we are not certain if it existed in 1922 because it was not registered in PNS.

— Murata-Sawayanagi, Nanako ; 澤柳, 奈々子: "The Memory in a Crisis: A Japanese Ship Helping Out Greek Refugees on the Quay of Smyrna in 1922", Mediterranean world = 地中海論集, Vol. 23, 2017 pp. 135–146. (hdl, PDF)

This earlier —possible— confusion over ship names seems resolved by now, as the research is accepted by the Benaki Museum in Athens:

Dr. Murata, Vice-President of Toyo University in Tokyo, specializes in modern Greek history and the history and culture of the Eastern Mediterranean. In her lecture "The Japanese Ship in Smyrna in 1922: The Narrative and the Story" she presents the findings of her yearlong personal research on the Japanese ship Tokei Maru. It is believed that this ship had rescued hundreds of Greeks and Armenians during the Asia Minor Catastrophe.

The animated short film "Tokei Maru" (2017, 20') - created by Zachos Samoladas, director/producer - is based on Dr. Murata's research. The film has participated in over 30 film festivals and has been honored many times. Mr. Samoladas will preface the screening.

— TOKEIMARU 1922-2022. Commemorating 100 Years since a Japanese Ship's Voyage to Smyrna FRI 23/09/2022

As s sidenote, it is also now commemorated in this name:

"Include School for All: Tokei Maru" in Thessaloniki.