There may be at least two different reasons for this difference. (I intentionally will not discuss which of them is applicable to this particular setup, but will present two reasons that may be at work in many similar situations.)

The first reason is that you consider only the stabilized part of the movement, while omitting the "[t]he bob of the pendulum will initially show some erratic motion" part. It may be that the resulting stabilized motion depends on the nature of suspension point movement during this phase. 

You say that "[y]ou can try to perform this experiment practically by attaching some heavy weight to the end of a rope and then revolving it by hand", I presume that you mean that it is possible to intentionally get both configurations, and probably, with some experience you will learn how exactly you need to start rotation with your hand to get a selected configuration. Then this "how exactly you need to start rotation" will represent exactly the difference that you are looking for. You will have two different ways of moving your hand, and two different outcomes, and you will be able to intentionally choose one of them.

So the first reason is that there may be some observable differences in suspension point movement during initial stage that results in different configuration. (In particular, the acceleration of the suspension point when it moves from rest to circular movement may be important.)

Of course, this leave open the question "what will happen if the suspension points moves in an ideal circular trajectory". Probably, only one specific configuration can be achieved, but to understand which one you will need to analyze this starting period. Nor your analysis (that looks only at stabilized movement), nor your experiment with hand (where you don't have precise control on suspension point movement) can be used to answer this question.

A simple example to illustrate this: consider an ideal sphere on an ideal horizontal surface. Let's mark some point on a sphere, and then push the sphere to start rolling. The sphere will stop at some point due to friction; let's ask: where the marked point will be on the sphere when it stops? 

We can do a stationary analysis, but (assuming the mark itself is of zero weight) we will not get any definite answer; any orientation of the sphere will be a possible stationary solution. But if we analyze the movement and the initial position of the mark, then it will be rather easy to find where the mark will be located at the end.

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And the second reason is [deterministic chaos][1]. In some systems, even minor perturbances in the initial conditions and/or external conditions during system motion can result in radically different results. In this case, the choice of specific configuration of resulting movement will depend on minor perturbances in initial state of the pendulum, or on minor external influences (wind, etc.) that we usually neglect in our analysis. So if you know the initial conditions and external influences with a very very very good precision, you know the result, but if you don't know them to a needed very high precision, the result can be seen as random.

Like above, this leaves open the question "what will happen in an ideal situation", but the difference from the first reason described above is that this question becomes unpractical in this case, because you never have an absolutely precise situation.

This is very similar to the flipping coin problem. While the movement of the coin can be seen as absolutely deterministic, still the outcome can not be predicted in a reasonable way, because small alterations in the initial state can grow large enough to change the result.


  [1]: https://en.wikipedia.org/wiki/Chaos_theory