A "conceptually"-deterministic signal might be a signal that is defined by an exact equation (e.g. sin(t), for all real numbers t), which has no random/stochastic/probabilistic component.
IF I am using the correct definition of "deterministic" (i.e. "not stochastic"), then I believe that all "real world" signals would not be deterministic, and would be stochastic (i.e. random).
A good example of a random signal is noise, such as the "static" noise that comes from the speaker of an AM radio when the receiver is not tuned to any station.
But music and speech would also both be considered to be stochastic, if you don't know what they will be, in advance.
And I guess you could just define random_signal_example(t) = true_random(t)!
Anything that you don't know in advance, the value of which is a random variable at any instant, is "stochastic"; i.e. a "random process".