This answer uses kind of a "proper less equal" symbol: $\lneq$
I would have expected that $<$ is sufficient, because it seems to be the same relation.
For $\subset$ vs $\subsetneq$ some authors interpret $\subset$ allowing self-inclusion $A \subset A$ to be true, while others do not, so the extra symbol offers less ambiguity for more ink.
What is this typically used for?
Update: We have $\ngeq$ too. :-)
It's not so much about being a different relation. Both $\lt$ and $\lneq$ mean the same thing. The second one is emphasizing that it is not equal.
In the answer, he emphasizes that it's not equal.
$\subset$ is sometimes taken to mean $\subseteq$, so if you want to be clear, you can write $\subsetneq$.
Yes $<$ and $\lneq$ are the same relation.
However, be careful about set inclusion: older texts, and some current mathematicians, use $\subset$ to mean $\subseteq$, and to indicate strict inclusion they might use $\subsetneq$ or $\subsetneqq$. Other authors who use both $\subset$ and $\subseteq$ will use $\subset$ to mean strict inclusion. These days, $\subset$ can be confusing. If you just jump into a book or paper and see $\subset$, it's best to check Section 0 or Chapter 0 to be sure what it means.
So $\subset$ and $\subseteq$ aren't necessarily different relations, universally, though they almost certainly are in a text that uses both. To avoid possible confusion, though at the cost of losing the nice analogy with $<$ and $\le$, it's probably best to stick to $\subseteq$ and $\subsetneq$.
Just a speculation. In case of partial ordering (for example, with respect to a cone) there are normally more situations than just $\ge$ and $>$. It may happen that a vector $a\ge 0$ (in the cone), $a\ne 0$ (not a vertex), but not yet $a>0$ (in the sense of the cone interior). It is where the demand for more notation symbols is most likely to come from.
