$X=Z^2$, $(n_1,n_2)<_{lex} (m_1,m_2)$ if $n_1<m_1$ or $[n_1=m_1$ and $n_2<m_2]$
I think not because its not reflexive. Is this correct?
Similarly, $X=R_{>0}$ (positive reals), $x ≪ y$ if and only if $\frac{y}{x} >10$. Again, this looks like a partial ordering, but I don't think its reflexive.
According to Wikipedia (and several books), there are two definitions of partial order, depending if it is strict or non-strict.
A non-strict partial order is a binary relation that is reflexive, transitive and antisymmetric. A strict partial order is a binary relation that is irreflexive and transitive (which implies asymmetry as well). Strict and non-strict partial orders are closely related:
If $<$ is a strict partial order, the non-strict partial order $\leq$ induced by $<$ is just the reflexive closure of $<$, i.e. $a \leq b$ iff $a < b$ or $a = b$;
If $\leq$ is a non-strict partial order, the strict partial order $<$ induced by $\leq$ is just the irreflexive kernel of $\leq$, i.e. $a < b$ iff $a \leq b$ and $a \neq b$.
Concerning your question, $<_{leq}$ and $≪$ are strict partial orders (and hence they are not non-strict since they are irreflexive).
Supplementary question: Can you show me a relation which is both reflexive and irreflexive?