I didn't do a good job when taking business law class few years ago, but one thing I learned is that when you make a counter offer, it automaticlly overrides the offer you received. The other party has every right to withdraw the offer.
Therefore, you need to be very careful and mindful of your bottom line and the risk to lose the offer when giving your conter offer. You should be ready to walk away when the offer is gone.
Of course, you can try to renegotiate like you did. However, if you back off from your position too quickly and too drasticlly, you'd be running the risk to damage your credibility. If I were the manager of this company, you had already left me an unfavourable impression - you are either being too greedy or you are just a bad negotiator.