Your Statefarm link is very informative and spells out very clearly the problems you may face and the possible solutions.
Germain to your original question is this paragraph which spells out the two, exterior vs. interior, and gives at least some preference for the internal (below slab)...
There are two basic types of drain systems for wet basements. One is a perimeter above-slab gutter system installed at the base of the exterior foundation walls on top of the floor slab. It doubles as a base material for the wall. The other type of drainage system is a below slab perimeter drainage system. The below slab system requires the partial removal of the concrete floor slab and installation of drainage pipe, making it more expensive than the base gutter system.
It is believed that an under-floor drainage system is better because the under-floor drains are believed to relieve the hydrostatic pressure before the water reaches the bottom of the floor slab.
I think the merits of one or the other are going to change depending on the house and it's particular circumstances. I think it is possible that the city system can back up and in this case it may be helpful to consult your neighbors. For instance, my basement floods but neither of my neighbors basements do. But if the groundwater is rising for me, then it will for them too. Another consideration is what type of building and how much below ground concrete does it have (possible water dam) down hill from you.
My point is, there are many considerations. A good company will garuntee and warrantee their work. I have seen Permadry come out and do addtional work for free becuase their proposed system did not work. You are on he right track and educating yourself and asking hard questions will ensure that your eventual provider will satisfy your demands. By the way, I am not employed by or shilling for Permadry. I only mention them becuase I have seen their work and I am sure there are others who will do just as good a job.
Finally, it is entirely possible to undermine a foundation when exposing one side of it or the other to install a drain system after the fact. The excavator will need to be carefull not the disurb the soil directly under the foundation / footer. However, there are likely to be voids and soft pathches develop under any old house as they did not pay as much attention to soil composition and compaction when most of these old homes were built and so that may not be the excavators fault and may in fact have been a problem before the drain system work exposed it or shined a light on it.
At some point, you may want to consider hiring a contractor to be your intercessor. The is what a contractor is supposed to do. The times that Permadry returned to do addtional work could very well have been because they knew that if they didn't they would not get our referrals any longer. I really can't say as I have not been with this contractor long enough to know the details of the situation.
Hope that helps.