I've previously written answers to similar questions like this.
That answer goes through and describes some of the differences with the terms you mention. Regarding the history of the term "American Indian," perhaps this answer of mine will provide some insight:
"American Indian" is a specified term that derives from the use of the term "Indian," which became the common nomenclature for the Indigenous Peoples of the Americas. It has now developed into a term specifically for Indigenous Peoples within the contiguous United States (Alaska Native is used for Indigenous Peoples in Alaska, Native Hawaiian for those in the Hawaiian Islands). This term has a particular distinction because it is cemented as a legal reference for Indigenous Peoples who have contracted agreements in the form of treaties with the United States government, which is the reason for its continued use. It is also a political term in that it designates an Indigenous Person/community/group as belonging to a polity--Tribes are nations.
"Native American" is a more socially acceptable term that arose during the 1960s and 1970s, primarily in academia, when postmodernist thinking and critical approaches became more prevalent in the academy. Though this is now a preferred term among some, it does not carry the same legal connotations associated with American Indian.
Regarding the term "First Peoples," Gregory Younging states:
This term is rarely used by Indigenous Peoples to describe themselves, although it is also not considered particularly offensive or problematic. It recognizes that Indigenous Peoples are distinct groups, without the political connotations of the term First Nations. It is not widely used in the literature, and some organizations have stopped use it, such as the Canada Council for the Arts.^1
In line with that, perhaps you might wanna know about the term "First Nations." This term is associated with Indigenous groups that reside within the boundaries of what is now known as Canada and is regularly used to make a distinction between Indigenous Peoples in Canada and those of us in the United States. Similarly to American Indian, "First Nations" has political connotations, though "Indian" has more legal bearing in Canada. First Nations gained popularity because Indigenous Peoples in Canada adopted the term in the 1970s. This was signified with it was adopted by the Assembly of First Nations, a national political organization.
"Indigenous Nations," which isn't necessarily a common term among Indigenous Peoples, is in reference to our identity as Indigenous groups and status as nations. "Indigenous" is quickly becoming a preferred term in many contexts, such as among the United Nations, as an appropriate collective term to refer to the Indigeneity of original inhabitants and preserves an element of cultural integrity. Combined with political verbiage, it references this cultural integrity alongside our political status.
Footnotes
[1] Younging, Gregory. Elements of indigenous style: A guide for writing by and about indigenous peoples. Brush Education, 2018, 64.