In Rudin's book, he defines an order $<$ on a set $S$ as one satisfying transitivity ($x < y$ and $y < z$ implies $x < z$) and trichotomy (exactly one of $x <y$, $y < x$, and $x = y$) holds. I was under the impression that this is a total order because every element is comparable and Rudin never made a distinction between a partial order and a total order. However, a set of lecture videos I'm watching build up a total order from a partial order, defining it as one satisfying reflexivity ($xRx$ for all $x$), transitivity ($xRy$ and $yRz$ implies $xRz$), and antisymmetry ($xRy$ and $yRx$ implies $x = y$).
I'm trying to reconcile these two definitions in the context of the standard order $<$ on $\mathbb{R}$. The trichotomy condition seems to be close to antisymmetry, but only vacuously, because I can't have $x < y$ and $y < x$. If I "expand" $<$ to $\leq$, then I know $x \leq y$ and $y \leq x$ implies $x = y$. Reflexivity is not satisfied in the case of $<$, but it is satisfied for $\leq$.
I can't say that $\leq$ is a total order because it fails trichotomy: I can't say exactly one of $x \leq y$, $y \leq x$, or $x = y$ holds because if $x = y$, then all three hold. I did some more reading and found there is a such thing as a "strict" total order, which seems to be a total order in the precise sense of Rudin, but without reflexivity.
With all of these said, there are two things I would really appreciate someone clearing up for me:
(a) Which of these definitions of total order are standard? Do we need to specialize to a strict total order in the sense of Rudin for the standard order on $\mathbb{R}$?
(b) How do I reconcile $<$ and $\leq$ in this case? When I define an order $<$, is there a sense in which I can "specialize" to $\leq$? I believe $\leq$ is usually a partial order, but it is not a total order because of trichotomy.