I wouldn't trust a random individual on the internet to correctly identify bearing walls. There are general guidelines for identifying bearing walls, but many exceptions to the rules.
Get a competent person in your kitchen to look at the structure. From there, you can figure out the engineering.
You say this house "has a truss roof". If the house truly has a truss roof, the only load bearing wall is at each end of the truss. But you contradict yourself when you say “the ceiling joists meet about 12 inches to one side of the wall”. The bottom cord of a truss (what the ceiling dry wall attaches to on truss construction) would be either one piece or spliced. They also would likely be 2x4 construction on a house that small. When you say the ceiling joists meet about 12 inches from the wall, I assume you mean that they overlap each other at that point. I would dearly hope that both ceiling joists reach all the way to the top plate of the wall in question. If not, and the fact that you say the wall is not directly supported to the building foundation, it looks like poor construction practice and not meeting whatever code is in place at your location.
I just finished removing a wall very similar to what you have described. But my wall was directly in the center of the house, parallel with the roof peak, and directly over the center beam of the house. My wife wanted a smooth ceiling, (no header), so the support had to be done in the attic. I ran the calculations for a 22’ span and found out that a span that long is totally impractical. So I put a column at 13’ from one end, which I used as a pipe chase for the kitchen sink vent. With only a 13' span, I was able to use two, 1¾” by 11¼ “ LVLs in each direction from the new column that I shimmed so they were a little more than ¾” apart. Each other end was in a wall so I was able to add 2x4s as a support and block the space between the basement beam and the subfloor so that the load was transferred down to the foundation. I also had to check that the resulting point load from these new columns did not overload the basement beam. I then fabricated some steel “hangers” from 4” channel and ¾” threaded rod to catch the two ceiling joists that overlapped the old wall. The two lapped ceiling joists nested perfectly in the 4" channel, and the threaded rod slipped between the two LVLs to above a top ¼” plate that spanned the twin LVLs allowed a firm bolted connection. The ceiling joists were shimmed as required to keep the ceiling level.
So I agree with Jimmy Fix-it on the inadequacy of a 4"x8" for a 13’ span. You don’t say where you are, north or south. If you are south, you don’t have a huge snow load to worry about. But I would spend the money and engage an engineer to evaluate the current design situation, as any changes you do could exasperate the problem.
Best Answer
It can be kind of hard to tell from photos.
Your house plans (blueprints) would tell you for sure, presuming the house was built faithfully to the plans. You really should have a good look at the plans, or get somebody knowledgeable to look at this in person. Or both.
Do you know which direction the joists are running? Floor and ceiling joists will be perpendicular to the load bearing walls (or beams!), not parallel to them. If the joists pass across the top of the wall (or beam) unbroken, then the wall is possibly not load-bearing, depending on your joists, the length of the span after you remove the wall, and the load on the floor above.
If the ends of the joists are resting on the wall or beam, then it is definitely load-bearing.
if there isn't another floor above this one, and you have engineered, prefabricated trusses holding up your roof, this won't be load-bearing. The trusses are engineered to handle the load of the roof by themselves. If they aren't engineered trusses, that doesn't apply.
Is the long common wall of the living room/dining room/kitchen directly on top of (parallel with) the long wall downstairs, and is the opening between the dining and living rooms perpendicular to that long wall downstairs? It looks like the photos might be from different directions? If so, then the common wall with the kitchen is probably the load-bearing wall, and the opening between the living and dining rooms may or may not be, depending on how the joists are laid out.
But you really should look at the plans or get somebody knowledgeable to take a look in person.