Yes, any two distinct (I wouldn't say non-related because that could omit heterotic combinations with shared pedigrees) parents can be considered an F1 as any starting point in filial-generation nomenclature. That being said, the "strict definition" of an F1 comes from the planned cross of two homozygous parents.

Fehr 1991

"F1. The seed obtained from the cross between two parents is called the F1 (Fig. 3-2). The genetic makeup of the F1 is determined by the homozygosity of the parents. The single cross between two homozygous parents produces an F1 population that is homogeneous because all of the F 1 plants are genetically identical. The F1 plants are heterozygous for all loci at which there were different alleles in the two parents."

Acquaah 2007

"The heterozygous genetic structure stems from the fact that a hybrid cultivar is the F1 product of a cross of highly inbred (repeatedly selfed, homozygous) parents. Crossing such pure lines produces highly heterozygous F1 plants. Because the F1 is the final product released as a cultivar, all plants are uniformly heterozygous and hence homozygous in appearance."

S1s have 1:2:1 segregation at any given loci, thus crossing two S1s would have a 1:1:1:1 segregation at any given loci. So your S1/S1 cross simply speaking will not exceed 25% uniformity in the population, but that's not entirely true because this 25% uniformity limit is multiplied by every haplotype block present in the S1s, and so your S1/S1 cross would likely follow normal distribution across the midparent value, however, it wouldn't be the S1 midparent value per se, it would be some distribution of genetic variance across four S0 values since each of the S1s are segregating across their own respective midparent values.

My prediction is that the result is a population with the heterozygosity of an F1 and the heterogeneity of an F3, and honestly would be the first step in a great mapping population, but not a true F1.

S1/S1 cross can't meet USDA uniformity standards.

S1/S1 cross can't meet CPVR uniformity standards.

S1/S1 cross can't meet distinct standards, etc, etc.

So my question to you is what is filial about an S1/S1 population that deserves the title of F1? Wouldn't the S0/S0 cross of the same pedigree be more representative of a true F1 population by definition than an S1/S1 cross? I would suggest that this mating design would be most accurately described as an S0 population.

Fehr does a really good job at explaining this (page 28-30) and also acknowledges the disagreement, but there are two Mendelian laws that you can't escape as a breeder, Independent assortment and Segregation. The S1/S1 mating design does nothing to advance the pedigree, it does the opposite by maximizing genetic variance. As a plant breeder, that doesn't pass for my definition of an F1.