The short version of the answer to you question is that Samuel Nicholas was the first person to Command the USMC, and therefore considered to be traditionally the First Commandant, while William W. Burrow was the first to officially hold the title and rank by law. And the reason it is different is tradition.

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The long answer is much more involved with Marine Corps History, Tradition, and Lore. While the Marines tout that they were first created in 1775 at Tun Tavern, the first Americans to hold the Title of Marine first took on the Role during the War of Jenkin's ear. They were, however, Royal Marines and the Corps doesn't recognize them as part of their history. These Marines are important in understanding the Marines long obsession with history and Tradition, as it is from the Royal Marines that symbols important to the USMC are based, including the now iconic Eagle, Globe and Anchor (EGA). After the founding of the Colonial Marines, and the United States Marine Corps afterwards, the USMC was barely more than an auxiliary to the United States Navy, mostly acting as security aboard ship and in port. They saw major combat during the Barbary wars and some of the Civil war, but between the two and during the War of 1812 they didn't have a very large role to play. After the Civil war, as Naval tactics and strategy began to change, the need for a Marine Corps was called into question, and the staff at the time needed to justify their importance. It was from this need that much of the amphibious landing doctrine that was so heavily used during the second world war first started to see development. This need for justification is also were the importance of tradition began. "Semper Fidelius" was first created, The Marine Hymn written, and much of what is now accepted as historic fact about the Marine Corps was first created. Some examples of this fabricated history is the Quatrefoil on Officer's covers (reportedly to help sharpshooters identify friendly officers, actually just the fashion trend at the time of creation), the blood stripe on the pants of dress blues (Reportedly to commemorate the dead during the Mexican American war, was actually the Army they are commemorating as Marines stayed on the ships), or the term Leatherneck (reportedly from the fact that the leather collars they wore at the time were to stop swords, actually completely for appearance). But while much of this was fabricated, it created an endearing and American identity, which helped them in the argument for existence, and now the Marine Corps holds onto tradition with an iron grip.

So that answers why the different name, and why it has stayed different. However, I think it is important to point out that the other Chiefs of Staff are not all named the same either. The Army has "The Chief of Staff of the Arm" and the Air Force has "The Chief of Staff of the Air Force", but The Navy has the "Chief of Naval Operations" and the Space Force has the "Chief of Space Operations". Notably, both the National Guard and the Coast Guard also have a Chief of staff, although only the "Chief of the National Guard Bureau" is in the Joint Chiefs of Staff as the Coast Guard falls currently under the Department of Homeland Security. You may find it interesting that the Coast Guard Chief of Staff is called the "Commandant of the Coast Guard". I don't know why.

The service chiefs (for the Navy and Army) were established during WW2. At that time there was the War Department (Army) and Department of the Navy. The Marines had the Commandant but did not have their own department because they were (and still are) part of the Department of the Navy. The service chief role was not well defined but generally was meant to facilitate planning and cooperation between the two services and to advise the respective service secretaries, who are civilian political appointees.

The National Security Act of 1947 created the Dept of Defense, with the Depts of the Navy, Army, and Air Force underneath. It also created the Joint Chiefs of Staff (JCS). The Commandant was an advisor, not a member, of the JCS until the 1970s. The title Commandant was kept for historical purposes but is still the equivalent of a service chief.