This is interesting. The technical guidance definitely says SMP is for employees, and mentions continuous employment, and employer, several times. You are right it doesn't specifically say that casual workers are excluded.
Then in the link you previously posted, the one about how different employment types affect it, there is a section on "Casual and short contract employees" which the guidance defines as follows: "A casual employee is usually someone who works for an employer, as and when they are required on a series of short contracts of employment with that person"
True casual workers, with worker status rather than employment status, with no mutuality of obligation and a genuine casual agreement, aren't employees and don't work on short 'contracts of employment'.
But that doesn't exactly answer the question does it? So I don't know! I don't think it's correct to say it literally makes no difference, I think if a genuinely casual worker might be entitled to SMP it certainly won't be as clear cut as an employee who meets the 26 weeks continuous employment plus earnings requirement.
I would still be interested to know whether the OP is a genuinely casual worker or whether she is an employee who happens to have zero hours in her contract, and either way, her employer is not correct to say that zero hours= no SMP, clearly it's not that simple at all, whether it's zero hours employment or a casual ad hoc, work a few-hours-every-couple-of-months type arrangement.