I think Wallace (1968) developed the ideas of hard and soft selection, as they relate to genetic load. He further explains the concept in Wallace (1975). As the idea was his, I'd go with Wallace's definition over Whitlock's. I haven't had time to watch the Whitlock video to see if they are saying more or less the same thing overall.
Consider a hypothetical population of individuals, each with 1000 loci (I'm basing example and ideas on an example given by Ridley 2004, pgs. 162-167). Heterozygotes at each locus has the greatest fitness (heterozygote advantage). Homozygotes at any locus will have reduced fitness. Therefore, the fitness of each genotype is given as:
Even if every individual in the population at time zero was heterozygous at each locus, Mendelian inheritance ensures that no individuals will be heterozygous at all loci after the second generation. Assume that fitness reduction at each homozygous locus is just 1%. Given that well over half of the loci will be homozygous, fitness of those individuals will be reduced to 0. They will die before reproducing. This is hard selection. If hard selection of this magnitude occurred in a population, then extinction would occur. This is what Wallace (in your link) is saying. This mortality due to the genetic load is added to the normal background mortality.
Soft selection compares the fitness of an individual to the genotype with the greatest fitness in the population, even if that genotype does not the theoretical optimum fitness. Soft selection replaces some of the background mortality with selective mortality, so the population size is not decreased overall.
Ridley, M. 2004. Evolution. 3rd ed. Blackwell Publishing, Malden, Massachusetts, USA.
Wallace, B. 1968. Polymorphism, population size, and genetic load. pp. 87-108 in R.C. Lewontin, ed. Population biology and evolution. Syracuse University Press, Syracuse, New York, USA.
Wallace, B. 1975. Hard and soft selection revisited. Evolution 29: 465-473.