Can a Foucault pendulum really prove Earth is rotating? According to this article, a Foucault pendulum proved Earth was rotating. I'm not sure it really proved it.
If Earth weren't rotating and a Foucault pendulum started in a state with zero velocity, it would keep swinging back and forth along the same line. If at its highest point, it has a tiny velocity in a direction perpendicular to the direction to the lowest point on the pendulum, then maybe it would have such a tiny deviation from moving exactly back and forth that a human can't see that tiny deviation with their own eyes, but that deviation would result in a slow precession of the pendulum and a day is so long that the direction it's swinging in would rotate a significant amount.
If you have a system where a particle's acceleration is always equal to its displacement from a certain point multiplied by a negative constant, and it's not moving back and forth in a straight line, it will travel in an ellipse which does not precess at all.
I think the same is not true about a Foucault pendulum. Its acceleration doesn't vary linearly with its distance along the sphere of where it can go to the bottom and the sphere of where it can go doesn't have Euclidean geometry. My question is can we really conclude from looking at a Foucault pendulum and the laws of physics that Earth is rotating?
Maybe if its initial velocity is controlled to be very close to zero, we can tell from its precession that Earth is rotating. Also maybe if its arc is very small, the rate of precession for any deviation from going back and forth that's undetectably small to the human eye will be so much slower than the rotation of Earth that we can tell from watching it that Earth is rotating. That still might not prove Earth is rotating, because if the pendulum has a tiny charge in the presence of a weak magnetic field, the magnetic field could also cause a slow precession.
 A: If at it's highest point, it has a tiny velocity in a direction perpendicular to the direction to the lowest point on the pendulum, then maybe it would have such a tiny deviation from moving exactly back and forth that a human can't see that tiny deviation with their own eyes but that deviation would result in a slow precession of the pendulum and a day is so long that the direction it's swinging would rotate a significant amount.
That is incorrect. A perpendicular component to the velocity would result in an elliptical movement around the lowest point(when seen from above). It would not result in anything like precession.
A: About the construction of a Foucault pendulum:  
I have read several accounts from teams that had constructed a Foucault pendulum. And indeed a recurring theme is that is very hard to get the parasitic effects down to a level where the Foucault effect dominates. Many a time a team saw with elation how their pendulum finally showed precession, only to realize that it was in the wrong direction. Also it is common to add a driving mechanism, so that the swing doesn't decay. But it's very hard to eliminate a bias from the driving mechanism. I read an account that went something like this. "We tweaked our setup until we obtained the theoretical precession rate. To be honest, we can't be sure whether our pendulum is doing a true Foucault precession, or whether we've merely dialed in the precession rate."
So, yeah: from a purely scientific point of view a Foucault pendulum setup is not a particularly good way to demonstrate that the Earth is rotating. (However, Foucault's gyroscope was, I'll get to that in a few paragraphs.)
The Foucault pendulum was the first setup that demonstrates the Earth's rotation without any astronomical observation. You're not looking outside, you're looking inside, yet you can observe the Earth's rotation. 
The Foucault pendulum is so evocative because the appearance of it is so simple. 
One or two years after the construction of the Foucault pendulum Foucault devised another way of seeing the Earth's rotation, while looking inside. Foucault commissioned his instrument maker to construct a device that Foucault called "gyroscope".
Froment, the instrument maker, succeeded in constructing a gimbal support with so little friction that deviation of the spinning gyroscope wheel from its initial direction was negligable. Also, the gyroscope wheel could be lifted out of the gimbal support, turned around 180 degrees, and reinserted. In both orientations the gyroscope showed the same direction of Earth rotation.
A: There is a 1969 paper by E. O. Schulz-DuBois titled, 'Foucault pendulum Experiment by Kamerlingh Onnes and Degenerate Perturbation Theory', discussing the research into pendulum swing by H. Kamerling Onnes (the Kamerlingh Onnes who is known for his success at cooling down Helium until it liquified.)
I have to say I don't fully understand the content of that paper, but hopefully I can report some of the findings correctly. If the subject is of interest to you: regard my reporting as starting point for further digging.  
Kamerling Onnes studied a pendulum of the following form: a rigid rod, with a double knife edge suspension providing the required freedom to swing.
Among other things Schulz-Dubois points out that for a conical pendulum with ideal performence there are two circular swing modes with the same period: swinging in a circle clockwise and swinging in a circle counterclockwise. Linear swing can be represented mathematically as a superposition of those two oscillation modes.
Let me introduce a new expression: zonal Foucault pendulum. A zonal Foucault pendulum is one that is not on either of the poles, nor on the equator, but somewhere in between.
In the case of a zonal Foucault pendulum the period of the two circular swing modes is not the same. The veering of the plane of swing can be represented as a beat frequency of the two circular swing modes.
Let me go over a general characteristic of a Foucault pendulum setup. What Foucault foresaw (and what others had missed) is that although the effect of the Earth's rotation is extremely small, in a Foucault pendulum setup it shows up because it accumulates. The problem is: the tiniest of imperfection can throw the demonstration off because the imperfection's effect also accumulates.
If a conical pendulum is not constructed perfectly symmetrical the natural period of oscillation is not the same for all orientations of the plane of swing. The difference will be exceedingly small, but as mentioned in the previous paragraph, the effect accumulates.
Schulz-DuBois writes about the solutions to the equation of motion for the asymmetrical case: "the first and second eigenfunctions describe ellipses with interchanged major and minor axes, and with opposite sense of circulation. [...] If the pendulum is excited to an orbit described by an eigenfunction if will continue to move in this orbit without change [...]. Any other excitation involves both eigenfunctions. Due to the frequency difference between them the pattern of pendulum motion changes with time." 
So: the problem of imperfect construction is acute.
(For completeness:
Kamerling Onnes had designed his setup in such a way that it was adjustable. He devised procedures to systematically home in on adjustments to eliminate any asymmetry. Schulz-DuBois writes: 'When adjusted his pendulum performed as expected from unsophisticated theory.')

More general discussion:
The question raised is: is a Foucault pendulum actually as good a demonstration of the Earth's rotation as the textbooks claim it is?  
Well, for sure the 1851 Foucault pendulum was. To this day it's the longest wire Foucault pendulum ever: 67 meters, and as we know: the longer the pendulum the less susceptible it is to imperfections in the construction.
The thing is, we should think in terms of having to shoulder the burden of evidence. Can you offer the veering of the plane of swing as evidence that is strong enough to hold up in court? 
I'm wary of pendulums that are driven. How can you be sure that the driving mechanism isn't introducing a bias? 
In the case of the Foucault pendulum there is much more than meets they eye. 
A: The thing that Foucault did was not just to predict that a pendulum would undergo precession, but also to predict which way the precession would go, and how rapidly, depending on the latitude of the observer.
(Although Cleonis points out in a comment and in another answer that ascribing all of this analysis to Foucault is a historical oversimplification.)
If you imagine a Foucault pendulum set up at one of the poles, with the Earth rotating underneath it, you should be able to convince yourself that an observer standing on the Earth would see the pendulum complete one precession cycle every day. Likewise a pendulum swinging in the plane of the equator would have no tendency to precess, and one at the other pole would (relative to the ground) precess the other way. The precession period works out to be $\rm1\,day/\sin(latitude)$, which at Paris is about thirty-two hours.
Many of the mechanical details of a Foucault pendulum are set up to reduce the contribution of the parasitic effects that you're thinking of. Foucault used a very large mass, to reduce the acceleration imparted by stray air currents, on a very long tether, so that individual swings of the pendulum are very slow and any intrinsic curvature or ellipticity is observable. He was careful that the material of the tether shouldn't "unwind" the way some multi-fiber ropes do, which would exert some extra torsion on the motion. And the pendulum was released by tying it to a horizontal fiber and then burning that fiber with a candle, rather than by cutting the fiber with a knife or having a person just give the pendulum a shove, to minimize exactly the sort of parasitic horizontal forces you're asking about.
Even then you might be able to explain away a single demonstration as a lucky fluke. The real strength of the Foucault pendulum comes when you repeat the experiment so that all these tiny parasitic forces ought to be randomly different, but the precession period turns out to be exactly the same, and then you repeat it in a city at a different latitude and the precession period is different by the right amount.
