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Please note that there is no direct relationship between heat capacity and rate of heat loss to the surrounding.
As I havbe mentioned, why a metal cup has a higher rate of heat loss is because of its shiny surface, but not its small heat capacity.
Lowering the heat capacity is not necessarily changing the material being used, like she can use a smaller beaker.
In this case, same material but smaller contact surface area with air-->smaller heat lost to the sorrounding, but it has a smaller heat capacity as its mass decreases.
You are using a wrong concept to derive a wrong conclusion.
Also, we indeed have a practice to ignore the time for measuring the temperature, unless it is specified by the question.