A spring is basically just a weak/hollow area that lets the water rise to the surface by the pressure of the water under ground. Side of a hill spring is different of course many times and it's often just a leak as the water tries to run from the top downhill through the hill and flows out the weakest area.
Digging around with a shovel should not normally hurt the average spring. Running large equipment could pack it closed tight or heavy vibration could collaps/pack the ground the spring runs though. In such a case packing an area is like shutting off a valve of course and the water stops. It could just not have enough pressure to continue beyond that point and stay under ground, or it could be like a fractured pipe and look for a new weakest spot to burst and run out. That could be anywhere! The path of least resistance, same as electric pretty much.
I have seen alot done with springs that had no effect like just digging a bit with a shovel and setting a tank to catch the water, then pump elsewhere. I know a house where they dug a large cistern by hand just aways from the spring then a small shallow ditch to the puddle/stream of the spring, filled that with gravel and let it flow to the cistern. Pumped to the house like any other water, but used a seperate deep well for drinking and cooking water. Spring water for about everything else, baths, clothes, etc.. and since this was open surface water they did dimp a little clorox into the cistern to keep it clean.
I also saw people with a beuatiful spring drill a deep well on the other side of the house about 300' away from the spring and it kiiled the spring! Once that water goes under ground you never know where it came from or where it actually flows. They drilled right through the path of the spring and dumped it into the well, not what they wanted to do!
Each spring is is different really, and the only real thing to do is be carefull with it and watch it as you work. If you see it slowing, STOP! If it is still flowing it may wash the opening back open and return to full flow.
Try to get a good idea where it is flowing from if you can, but that is not easy. Kinda like a pipe in a giant mound of dirt. Toss a flexable tubbing in the air and 5 tons of dirt, just because the fawcet is on the north side does not give you any idea where the tubbing is laying in that mound! Same with the spring, many actaully run up hill! I have seen a few people think the spring was from up hill and dig right into it in the downhill side. Water does not flow down hill when under a presure, it flows anywhere of least resistance. So if the ground is very hard near the surface down hill, but there are under ground fractures running up to softer ground and enough water presure at the source that water runs up hill under ground same as if in a pipe. Then once it breaks to the surface of course it will run downhill, but inside that hill it could run anywhere.
"One idea I had was to gently dig out the spring about 16" to remove any leaves and silt. Then cut the ends off of a 50 gallon barrel and set in the spring and back fill the exterior of the barrel wall with clay and the inside with fist sized rocks back to the previous ground level. Then I could place a pipe on the side of the barrel to move the water to wherever I needed it. Some of the questions I have would be how to keep the water inside the barrel, or the majority of it anyway and keep it from seeping around the bottom."
Gently is good, but why do you want to cut out the bottom of the barrel, leave it in and that's how you stop the water for runing out. Use a teflon lined barrel like for corn oils, it should never rust out from the inside. You can put rocks in the barrel, I might leave it empty myself just depends, be careful back filling with clay. I would just dig the hole the size of the barrel and toss dirt in around it loosely, let it settle in on it's own.
