Question: > 
> **A bucket of water is** tied to a rope and **swung in a vertical circle**.
>  The distance from the bucket centre to the axis of rotation is
>  $2.08m$. **Calculate the angular velocity** (in $rad s^{-1}$) **of the
>  swing**, to prevent the water from falling out of the bucket. *[emphasis mine]*
> 
> 
> 

My answer $\sqrt{5g/l} = 4.856$ refers to the minimum initial angular velocity at the base of the circle (such that the water will remain in the bucket even when it reaches the top of the circle), while the marking scheme of the test which this problem came from has $\sqrt{g/l} = 2.172$ instead, referring to the angular velocity at the top of the circle.

I was marked wrong for this problem, and my lecturer tells me that I have over-complicated the problem. (The disagreement is basically over the interpretation of the problem, specifically its last sentence.) So learned ones, I seek your input: was I right to supply the velocity at the start of the motion, or was I over-complicating the problem?

Answer: 
I probably would have calculated the angular velocity at the top of the circle. But the question is clearly ambiguous, and I don't think it was unreasonable of you to take "the angular velocity of ... the swing" to mean the *initial* velocity of the swing, starting from the bottom. The question should have said "calculate the angular velocity at the top of the swing", in order to make it clear what's being asked for. Most likely the person setting the question didn't actually consider that the angular velocity would be changing throughout the bucket's motion. I've never marked a test, but if you started by calculating the angular velocity at the top of the swing and wrote down $\sqrt{g/l}$ then I probably would have given you the marks for that (since it's the correct answer according to the answer scheme) and just ignored the extra part.
At the base of the circle, the water stays in the bucket without any extra work. The reason is that it is attracted to the Earth by its gravity. Try to get some water into a mug and you may experimentally observe that it will stay there if the mug is oriented in the usual "breakfast mode". (If it is oriented in the opposite way, the water can't stay in the swinging bucket whatever its acceleration is.)

The relevant point where it's hard to keep the water in the bucket is clearly the top of the circle. At that point, the water is accelerated towards the Earth i.e. out of the bucket by the acceleration $g$. One needs to compensate it by the centrifugal acceleration $l\omega^2$. So the minimal value of $\omega$ that keeps the water in the bucket is
$$ l\omega^2 = g,\qquad \omega=\sqrt{\frac gl} \approx 2.17\,\,{\rm rad/s}$$
As you can see, I would also give you 0 points because you seem to be confused about the direction of the gravitational force or the direction of the centrifugal force which is a pretty serious bug. The only conceivable thing you could get a fraction of the credit for is the fact that your answer was dimensionally correct. Aside from that, I can't imagine a justification of a nonzero credit for your answer that couldn't be used to reward *any* wrong answer. This was really an easy question and your doing complicated things that lead you to the square roots of five indicates that you don't understand these simple things.

BTW what users may post on Stack Exchange are "questions", i.e. sentences terminated with a question mark whose purpose is to learn the right answer out of a larger of set of a priori conceivable candidate answers. The purpose of this server is not to "verify" things, especially not someone's predetermined wrong propositions. Your expectation that we would verify that your lecturer was wrong wasn't rationally justified, I think. Even statistically, I am confident that in most of such situations, the students are wrong and the lecturers are right. At any rate, your example belongs to this category.