The definition of a retaining wall is quite simple, it is a wall that retains solid material. The material is typically soil, but could be rock, grain, or any other solid that cannot hold itself up. If the retaining wall holds back water, it is a dam. By this definition, a basement wall that has an unbalanced soil load (essentially 100 percent of basement walls) is a retaining wall. It is also a foundation wall.
The confusion seems to occur when discussing the various types of retaining walls. A gravity retaining wall acquires its strength from the combination of the weight of the pieces and friction. Typical gravity walls are made of stone, concrete block, gabions etc.
A segmental block retaining wall gets its strength primarily from reinforcing installed into the soil behind the block, which is not typically necessary if the wall is quite low.
A cantilever wall acquires its tensile strength from reinforcing steel placed in the concrete (most cantilever walls are reinforced concrete) and the compressive strength of the concrete. A cast in place concrete basement wall supported at the top by floor joists is analyzed as a diaphragm rather than a cantilever wall. In this way, a house basement wall is similar to a tied back sheet pile wall, which is another type of retaining wall.
There are also semigravity walls, which derive part of their strength from dead weight, and part from the strength of the concrete and steel reinforcement.
There are many other types of retaining walls, including soldier pile and lagging, caisson walls, various forms of sheet pile walls, secant walls, soil cement walls, and slurry walls. If there is an unbalanced load on the wall, it is a retaining wall, regardless of external supports.
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