$\subset$ vs $\subseteq$ when *not* referring to strict inclusion

Different people use different conventions. Some people use $\subset$ for proper subsets and $\subseteq$ for possible equality. Some people use $\subset$ for any subset and $\subsetneq$ for proper subsets. Some people use $\subset$ for everything, but explicitly say "strictly proper" in words when they feel it matters. I do not believe that there is a consensus for the meaning of $\subset$. My own personal advice is to use $\subseteq$ and $\subsetneq$ when you care to be precise, and $\subset$ when you are feeling lazy.

This is a very troubling issue with notations - it might not be uniform.

In many places $\subset$ implies a proper subset, while $\subseteq$ implies a possibly improper subset. In books you will find the definition somewhere, but in questions here... you just have to "guess" the right definition from the context.

Personally I am always in favor of clarity (when possible), $\subseteq$ and $\subsetneq$ are my choice of symbols. One of my teachers even takes $\subseteqq$ and $\subsetneqq$ for even greater clarity.

That depends on the author, see here.