You have conflicting needs: good tele lenses are expensive but, since you mention image quality as a requisite, cheaper lenses are obviously inferior when it comes to it.

Besides that, "saving" today by spending for a lens which will not maybe give you the right quality could be a false saving.

That said, I would consider the 70-200 f/4 L not IS. Keep in mind that if the subject moves IS is relatively useless, since it can only compensate (obviously) for camera shake. Depending on the kind of subject that you are interested, 200 could also be not enough: but going to longer focal lenghts while keeping a good quality is very much outside of your budget.

If you feel that the L lens is too much, among your options the 70-300 (which is also USM) would be arguably better given the greater flexibility in the range of focal lengths available.

PS you can find a review of this lens which mentions the same doubt that you have:

I reckon most shoppers in this range are torn between the EF 70-300mm f/4-5.6 USM IS and the EF-S 55-250mm f/4-5.6 IS - technically they're darn closed. So rest assured that you're not alone in your quest grin.