There is no "ambiguity" between space and space-time. They are clearly defined terms. Space means the 3d space that we experience around us. Space-time includes time as a fourth dimension.

While time as a dimenauon can be seen as a mathematical formality (it manifests as a different physical quantity in our observations), it does behave much (but not exactly) like the spatial dimensions.

The entirety of GR, especially the space curvature part is formulated without embedding space and time into a higher dimensional space of any kind. This avoids any speculation about extra dimensions "around" our universe. It also causes the enormous mathematical complexity of GR.

Back to the topic of gravitational waves: these arise out of Einsteins field equation which describes space-time curvature as a result of the distribution of mass and energy in space. These equations show the property that certain small deviations of curvature from their resting state propagate through space as waves. These curvature changes can be measured as changed in physical length of solid objects. This is what LIGO achieved. There is no room in current theories for any different kind of propagation medium for gravitational waves.

I seems to me that you have some difficulties understanding SR and GR. This is understandable because both describe phenomena that are not directly observable in our everyday lives.