In the final analysis, documentation is the key. It either makes or breaks liability issues. Obviously mistakes were made here and that's perfectly normal for any business, let alone ours. When you dropped the limb and dinged the pool, you did the right thing when you had a pool guy come out. At that point, a statement should have been drawn up, by you or your attorney, and signed by the pool guy wherein he affirmed that he examined the pool and found it free from any damage that would lead to later problems. In this way, you would get him involved in any legal action, at a later date, and spread the liability, lessening your portion of any actual recoverable damages. A ton of photos should have been taken, as well.
To sweeten the deal. if you could have gotten the home owner to sign off on a waiver of liability document that agreed to hold you blameless for any future damages that might occur to the pool, in return for your giving him some kind of monetary break in your final price on the tree job, you would be sitting fairly confortably, now. You might be surprised how many home owners would sign such a document for an immediate break on the price of a given gig. As long as the break was less than the current $1500, you would have done the right thing. Of course you might need a crystal ball to figure out the break amount, at the time of the damage.
A possible problem, here, assuming you didn't get a liability waiver, would be the qualifications of that first pool guy. That would be an issue in court, as opposed to the other pool guy who, down the road, dumped the fault of the leak on you. If the first pool guy was seen to be more qualified, and a jury/judge believed that the damage you did was not the cause of the leak, it might decide in your favor. Of course the opposite might happen if the second guy was seen as the expert.
A tangential issue would be the insurance coverage of the first pool guy. If his insurance came into play with his signing off on your initial damage, you might get the attorneys from that insurance company involved and get some free help, there. If your insurance company got involved, as well, a team approach might get you off the hook, so to speak. Keep in mind that different states have different laws regarding how this scenario might actually go down.
Once the leak was found, you should have immediately gone out with the first pool guy and, again, gotten him to sign a statement giving his opinion on what was going on, to the effect that the leak was not caused by the limb drop, some time back. At that point, the customer would reach the point that he would either sue you, or drop the ball and fix the leak himself.
Of course you do need to decide what the hassle factor is here, for you. With legal wranglings, your time might be worth a lot more than the $1500.
As to your offering him a settlement: Once you looked at the leak area, something I hope you have done, lately, you probably got a very good idea about whether or not you were the cause of the problem. While your business is not pools, if the leak is in the exact area where you dropped the limb, and any surrounding structural damage might be seen to manifest in an uneven distribution of the water pressure, leading to a leak, as a pool owner, myself, I have a pretty good feeling that you will know if you have a case in contesting the damage in court.
Best of luck to you.
HOLD ON!!! I just reread your first post. You say that the home owner had the first pool guy come out. That changes the story, possibly BIG TIME! You can consult an attorney, which you should definitely do, but my gut feeling is, now, that the first pool guy is a major part of this suit, whether he likes it or not. You trusted the home owner to get a fellow who knew his stuff, and you took the first pool guy at his word. You trusted his expert opinion. You did not feel compelled to hire your own expert, as the home owner went ahead and relieved you of your liability, here, by paying you and sending you on your way. If you had your own expert come out, and the courts will determine if that was needed, he might have pointed out things that needed to be done to prevent a leak, in the future. The fact that the home owner hired the pool guy and that guy assured both of you that everything was OK, is crucial in your attempting to seek a favorable ruling. Again, you did not have any reason, at the point where the first, home owner-hired, pool guy gave the OK, to go any further in preventing future damage. Had you done so, the leak might never have occured later down the road.
DON'T SETTLE HERE, WITHOUT CONSULTING YOUR ATTORNEY. Be sure to let him know that it was the home owner who hired the first pool guy and that the home owner was satisfied with the initial finding and then paid you, in full.
I have sued people, successfully. I have been writing contracts since I was twelve. My contracts have held up in court and I have been told, by lawyers, that I should teach contract law. I have an excellent "feel" for the law and I feel that you might not have any liability here, if this case is handled correctly. That being said, I am NOT a lawyer, so everything and anything I said, above, might be totally off-base and out in left field.
Again, good luck!