Getting Stung
Do you think there is a possibility that you were stung because you hired someone who was cheap/low cost and thus they didn't have the skills and experience necessary to create a winning design for you?
They do say pay cheap pay twice, do you want to repeat the mistake a third time? Your budget may be low, but you can't expect to pay peanuts and not get monkeys from that. That's the reality of the outcome of paying a lower price for a design service.
Good design doesn't have to be hideously expensive, but designers have bills to pay also and as a skilled profession (not everyone is a good designer, it's very much a skill that takes years to hone), you can't really get a decent designer for low prices.
If the price is low the corners will be cut somewhere ... the design will be not really good enough for something that could be your brand identity for many years to come (as you have found). Or perhaps it might look ok, but it might not be entirely original and has been based on a template so they can churn out designs at low cost ... but they aren't really original designs. You may find other companies with a similar design/same design.
Any professional designer will know that the advertising opportunity in this situation is close to zero I'm afraid and thus this doesn't make it worthwhile for them to cut their pricing in return for the good advertising opportunity.
Even if it said on the website 'logo designed by' and a link, and even if the traffic was huge (which I doubt it will be, and it would have to be really huge), the type of people visiting the site won't care who has designed the identity and aren't likely to click to find out.
The site traffic won't be mainly businesses owners needing design services, or startup businesses needing design services. i.e. they are not the designers target market/potential customer base.
To summarise, the minimum you can expect to pay to receive perhaps three concepts to chose from, and be reasonably assured that the designer is fairly good, is something like £200.00 and you'll be lucky to even find a good designer that only charges that much. Really a good designer is more likely to charge between £300 and £500 for three - five concepts.
Does you budget spring to anything like that?
Best Wishes
Amanda