In general I think the concept of weak categories is something worth keeping in mind when the new edition is done. I have earlier referred to Vroomfogles great work at reverse engineering the present costs. There was a couple of skill costs that was hard to replicate correctly without the skills being in different categories. Even if it is possible to work around those there might be other reasons why weak categories is wanted.
On the other hand I think it perfectly possible to set up the skills and categories so that strong categories work. If we take the example you are discussing in this thread then the real problem is that you are using a skill list that is flawed. The idea that a character might be expert in fine mechanics like locksmithing, but not being able to use his other skill in anything other is stupid. The correct skill list would have the skills lock lore and the skill fine mechanics. If the thief is great at lockpicking then he should also be able to use his skills for related crafts. The same applies to dealing with traps. To say that a character is unable to use wood tools for making a house, but a master on using wood tools to construct a trap is silly. On the reverse the lock picking lore knowledge would be essential for opening the lock in any timely fashion. The reason lockpicking can open the lock is because it is assumed some part of the lore about locks is gained through practice. Imagine a the master clock maker with a master skill in lock lore...would he really be unable to combine his two skills to solve the problem?
It is great when you can have a skill that include everything about a task, this is true for swimming and a number of other skills. The downside of enforcing a single skill for a task is that then you end with situations when the skills become so narrow that you need another skill just because it supposed to be taken by another professional. In many cases the system already has a lore skill and a "skill" skill, but the RM designers didn't follow things to their logical conclusion. Make a better skill list/ category list and the problem go away.
Looking at the starting questions
1) A group of skills that share a thematic concept?
This is true for those categories that have category progression. Need not to be true for categories that use the combined progression.
2) Is it a grouping you can learn together with category Ranks?
In some cases yes and in some cases no. RMSS shows with great success that there is need for both and that they can coexist with minimal bookkeeping.
3) is it skills that share the same cost?
For the first printing of RMSS that was not true. Yet, I do think all skills within a category should share the same cost.
4) is it skills that use the same stats (or 2 of the same stat with one differing?)
I think it is a good idea for the skills to share the same stats, but really this has nothing to do with the category concept. You can have skills share stats totally independent on categories.
If we sum it up...there are plenty of relationships between skills. In some cases the skills are very tightly connected to an archtype like thief, athlet, diplomat etc. those cases the category ranks are essential to model reality. Other types of relationships is more tied to them sharing stats. In yet other cases the skills are tied together better by belonging to training packages. You could have a set up when you get some extra benefit if you buy a certain set of skills (beastmaster spells can be taken during a lever if you also buy skills related to animal handling).
The idea that you need to nail down exactly what is meant by categories before discussion can happen is IMHO not correct. Communication is about understanding the context of what is said. The category mechanics will serve different purposes for different categories, I think the forum readers are well equipped to handle any ambiguity that is caused by this.