History:
Wikipedia's article Indian Territory says:
Indian Territory later came to refer to an unorganized territory whose
general borders were initially set by the Indian Intercourse Act of
1834, and was the successor to the remainder of the Missouri Territory
after Missouri received statehood. The borders of Indian Territory
were reduced in size as various Organic Acts were passed by Congress
to create incorporated territories of the United States. The 1907
Oklahoma Enabling Act created the single state of Oklahoma by
combining Oklahoma Territory and Indian Territory, ending the
existence of an unorganized unincorporated independent Indian
Territory as such. Before Oklahoma statehood, Indian Territory from
1890 onwards consisted of the Cherokee, Choctaw, Chickasaw, Creek and
Seminole tribes and their territorial holdings.
The Wikipedia article has extensive references to the acts and links to other articles on the Cherokee, Choctaw, Chickasaw, Creek and Seminole people.
The problem with Census information:
There are a couple of things to consider.
Except for the 1940 Census, where the enumerators were asked to mark the person in the household who was the informant, we don't have any information about who gave the answers to the census taker.
There's no guarantee that the enumerators followed the instructions correctly, but we can read the instructions to find out what they were supposed to do.
1900 CENSUS: INSTRUCTIONS TO ENUMERATORS
- Column 13. Place of birth of person.-The object of this question is to get the birthplace of very person living in your district. If
the persons was born in the United States, enter in column 13 the
state or territory (not city or town) of the United States in which he
was born. A person born in what is now West Virginia, North Dakota,
South Dakota, or Oklahoma should be reported as so born, although at
the time of his birth the particular region may have had a different
name.
Note that the instructions specifically list Oklahoma as one of the places where the enumerator is supposed to record the contemporary place name instead of the name the region had when the person was born.
We can guess about who gave the information to the enumerator, but we can be pretty sure
that the informant wasn't Effie's birth mother. It could have been the 2nd wife, or it could have even been a neighbor.
Place Names in Georgia:
The Geographic Names Information System (GNIS) datbase "holds the Federally recognized name of each feature and defines the feature location by state, county, USGS topographic map, and geographic coordinates."
Users can search by State and then County for all entries in the database, or by selected features. The screenshots below are for searches in Georgia for "Indian" and 'Populated Place' or 'Post Office' as the features.
There is one Post Office, Indian Springs.
You could try looking for the term in local histories, in gazeteers, in genealogical publications (e.g.by using PERSI, the Periodical Source Index) and other reference works. You might be able to find area diaries or ephemera by searching on ArchiveGrid or contacting the local historical societies or genealogical societies. Have you investigted the Enumerator (i.e. are they from Georgia, and if so, from which part of the state)? Have you looked in historical newspapers around that time for instances of "Indian Territory" referring to Georgia rather than Oklahoma?
Whatever you find, bear in mind that one record doesn't constitute proof about any fact or event. For proof, you need a well-argued proof statement, showing that your research has met the current genealogy standards. For births from the 1880s, there won't be statewide registration for either Georgia or Oklahoma, and you'll need to look for delayed birth records or to build a case from other records.
Resources:
