SNP is not a specific data type but rather a biological phenomenon. The abbreviation "Single Nucleotide Polymorphism" only means there is a variability (between individuals) in a single letter in a specific position in DNA sequence. It depends on the speaker how they decide to describe the variability; wether they choose to focus on the possible sequences and say for example "There is only A or T in this position" or they focus on possible human genotypes and spells out all combinations (AA, AT, TT).
What constitutes as SNP data depends entirely on research question, used organism (diploid/haploid/polyploid) and experimental design.
Article you link in your comment uses oat lines and not individuals from outbreeding population. Oat like arabidopsis is naturaly highly self polinating so creating highly homozygous inbred lines is relatively easy. In this case it is reasonable for the researches to expect only AA or aa genotypes in each line.
Edit: Since you asked for more examples where the researcher chooses to focus on alleles instead of genotypes. Many SNP genotypes distribution fit Hardy-Weinberg equations. If that is the case, the frequency of genotypes can be calculated from an allele frequency in population. The researcher might ask: "Is the frequency of allele a different in population/group 1 from population/group 2?" They collect data from indivuduals in group 1 (AA, AA, Aa, Aa, aa) and group 2 (Aa, Aa, aa, aa, aa). They decide to treat each allele as independent measurement (not dependent on individual it comes from) and poll the data: group 1 (AAAAAAaaaa), group 2 (AAaaaaaaaa) and then they do simple binomial test to answer their question.
Note: Even if Hardy-Weinberg equilibrium is not met, the questions about of allele frequency are is still valid biological questions (with slightly different use/interpretation down the line).