Is there a difference between ignorability and strong ignorability? The whole idea around ignorability is still leaving me a bit confused. I did read this post from the site: Strong ignorability: confusion on the relationship between outcomes and treatment. It had made things a bit clearer. But I'm trying to figure out what is the difference between ignorability and strong ignorability. It seems all the literature discusses strong ignorability as an assumption that we check in theory to determine the "quality" (there is probably a better word to describe it that is more poignant, but I'm not sure what) of our experiment, but I don't see anything on just ignorability.  Does it have a mathematical formulation like strong ignorability's $(Y_0,Y_1) \perp \!\!\! \perp  T|X$? This is all new to me and feeling a bit overwhelming....

Unless there is a specific reference in which you saw "ignorability" defined as something different from "strong ignorability," these terms are conventionally used interchangeably. Both are referring to the assumption that $(Y_0,Y_1) \perp \!\!\! \perp  T|X$.
For what it's worth, the term "unconfoundedness" is also typically used interchangeably with these terms: all three terms typically all refer to the same assumption that treatment is independent of potential outcomes (conditional on $X$).
