James, there's no general rule. The ASME/ANSI B.31 Code for Pressure Piping says nothing on minimum distances between welds in any of its sections.
What sometimes exist are companies requirements or recommendations. If you work for one of those companies, you should do what they ask for; after all you're paid for that.
What also exist are some magic numbers that nobody knows where they came from. For example, back in my days of erector engineer, many piping engineers and foremen said that the minimum distance between welds should be three pipe diameters, if at all possible. This means that if it isn't possible, well, it can be less than that.
Now, why three diameters and not two and one half? Because that's one of those magic numbers in engineering that nobody knows where they came from.
In your case, it's easy. Your client has specified the distance he wants and you should follow his requirement.
Giovanni S. Crisi
Sao Paulo - Brazil

My recommendations are similar to the ones you just summarized. I try to consider the NDT method to be used. If they need to do shear wave I try to keep the two welds spaced far enough apart so that the inspector has adequate room to slide the transducer back and forth as needed. I often get asked how close a fillet weld (like at the edge of a full encirclement reinforcement or repair band) can get to a nearby butt weld. If they can't keep it at least a few inches away I would rather see the fillet actually overlap the butt weld so you end up with something that resembles a full penetration butt wedl with reinforcing fillet (similar to welds that join a reinforcing pad to a pipe branch connection)