If the sole known function of a gene is to activate a transcription factor, would that gene also be considered a transcription factor, or is there a word for such genes that are further upstream on the transcription activation cascade?
Yes. For an example, see this list of targets of NF-kB (a transcription factor). Many other transcription factors are included there. As for a TF that does nothing except activate another, single TF? I don't know that those exist - TFs tend to modulate multiple genes.
From the wikipedia article on TFs:
In molecular biology and genetics, a transcription factor (sometimes called a sequence-specific DNA-binding factor) is a protein that binds to specific DNA sequences, thereby controlling the flow (or transcription) of genetic information from DNA to messenger RNA.
The nature of the gene affected is irrelevant, a protein is a transcription factor if it binds to a gene's promoter and regulates that gene's transcription. Whether the regulated gene also codes for a TF does not enter into it.
You need to re-write your question, it is ambiguous and your use of terms is incorrect... Assumption: by "activation" you mean "activation of transcription resulting in the expression of the transcription factor"
1) Transcription factors are proteins
2) Genes are comprised of DNA elements
A transcription factor can be involved in initiating the EXPRESSION of a transcription factor, whereafter that second distinct transcription factor initiates the EXPRESSION of another gene that encodes another transcription factor.
Answer: No, genes do not "activate" transcription factors*
*Unless you are proposing the philosophical question of whether the DNA binding domain itself, which endows the transcription factor with a state of being active duty (i.e. fulfilling its purpose as a transcription factor), and thus that purpose is fulfilled only when the DNA binds the the TF, then a DNA binding domain can indeed "activate" the TF... but I'm pretty sure this isn't what you're asking.