The answer is liquid cream. Milady states this directly, right alongside its description of soft soap: liquid cream soap is used in electric latherizers.
The book distinguishes it from soft soap, which it describes as available as shaving cream in a tube, jar, or press-button container instead — a different lathering method entirely. Electric latherizers are named as one of the most common, sanitary, and convenient lather-making appliances in a barbershop, but they specifically require the liquid cream form.
Milady Standard Barbering, 6th ed., Ch. 5 (Implements, Tools, and Equipment), p. 129: "Liquid cream soap is used in electric latherizers."
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