For the OP, the last cite seems to be a curious discussion, possibly where some participant were trying to add correct but inappropriate, and thus confusing, information to the question. As pointed out above - if you are using spherical coordinates the question of adding a change in location is trivial if the change is also in spherical coordinates.
Where it all goes messy is where the change is specified as a bearing and a distance, or some other mix of things that involve distance. If I say the target location is 10 miles at 130 degrees true, there are complicating matters (and implicit assumptions) that lead to problems.
If you take out a compass (and ignoring for the moment the difference between magnetic and true north) and you walk or fly a constant bearing, you will trace out a Rhum Line on the face of the Earth. Rhum Lines are obviously useful ways of navigating, as they are easy if you have a compass. Also, if you have a Mercator projection map, all straight lines on the map are Rhumb Lines. However, a Rhum Line is not the shortest distance between two points on a sphere, and does define a straight line on a sphere. A Great Circle is a straight line on a sphere. But a great circle will, in general, require a constantly varying bearing (relative to north) as it is traversed. But it also makes perfect sense (on the surface of a sphere) to define a delta in position as a distance along a great circle. This will define the shortest path to the target location. So defining a delta in position by bearing and distance is potentially ambiguous, and one where the error in position depends upon where you are on the Earth and what direction you are travelling. Which is why, if you can, one avoids doing so. However lots of real world navigation has dealt with such issues on a daily basis for centuries.
The other problem you get is that the Earth is not actually a sphere. The next useful approximation is that it is an oblate spheroid - it is fatter at the equator. Again, if you define position only using spherical coordinates you don’t have any problems. But the moment distance comes into play you yet another source of error to handle. In order to cope, there is an internationally recognised definition of the shape of the Earth, and how it is to be used for defining position. This is the WGS 84. GPS satellites have their position defined relative to WGS 84, and the positions a GPS receiver provides are thus rooted to this definition of the Earth’s shape.
It does however get slightly worse. If you want to use a map, you need to take into account how the Earth is projected onto the flat surface that is the map. This is a huge area, and there is no right answer. However one constant is that a sphere is a rotten thing for many survey tasks, and a lot of the time people really would prefer to work in a coordinate system that is rectilinear. On a small enough scale treating a section of the Earth as flat mostly works, and maps and measurements work out, even when the spherical shape of the Earth is ignored. But you need to define how that flattening is done. And because you can only get away with such flattening for a limited distance, you need lots of places where such flattening is defined. These are the local datums, and every country on the planet has them. Big countries, have quite a few. Where you get into real trouble is in moving from one datum to another, because you inevitably have discontinuities in your locations as you cross. But printed maps are mostly projected using these local datums. These datums provide a local rectilinear coordinate system. (The grid is defined as made up of “eastings” and “northings”, and can be on a feet or metre pitch) Many GPS receivers can automatically provide locations in these coordinates, and are programmed to convert from datum to datum as appropriate. Defining a delta position here is not a trivial issue. Within a single datum it does make sense to talk about x miles east, and y miles north as a delta. If the delta takes you into another datum it gets awful.
As much as you can you always want to work in spherical coordinates, but the grim reality is that there are many tasks for which this still isn’t the best, and coping with local datums and Mercator projections will remain.
For geocaching, it is possible that the issues involved could be enough to render the location of a cache in error enough to matter.