Definition of an Ordered Field A text I'm looking at has the following definition of an ordered field:  

DEFINITION A field ($F$, $+$, $\cdot$) is ordered iff there is a relation $\lt$ on $F$ such that for all $\quad\quad\quad\quad\quad x,y,z\in F,$  
(1) $x\not\lt x$ (irreflexivity)
  (2) if $x\lt y$ and $y\lt z$, then $x\lt z$ (transitivity)
  (3) either $x\lt y$, $x=y$, or $y\lt x$ (trichotomy)
  (4) if $x\lt y$, then $x+z\lt y+z$
  (5) if $x\lt y$ and $0\lt z$, then $x\cdot z\lt y\cdot z$  
$\quad$ Taken together, these properties ensure that the field elements are linearly arranged, and that the ordering is compatible with the operations of addition and multiplication.  

(Also, in another text that I'm using the definition appears with (1) omitted.)   
The text says that these properties ensure that the field elements are linearly arranged, which makes me think of the definition of a linear order (or a total order). According to the same text, a linear order is a partial ordering $R$ on a set $A$ with the property that every two elements are comparable. Thus, a linear order (which I'm assuming $\lt$ is for $F$ in the definition above) is one that is reflexive, antisymmetric, transitive, and also satisfies totality (or comparability).  
Assuming I'm correct that $\lt$ is a total order on $F$, then how does it meet the definition of a total order, when for instance it does not satisfy totality and isn't reflexive?
 A: Your order is reflexive by combining axioms (1) and (3). 
And isn't Axiom (3) totality?
Note that (1) can be omitted if "either" in Axiom 3 is replaced by "exactly one of" 
A: The text is defining a strict order, that is a relation that is irreflexive and transitive. In this case, totality is expressed by trichotomy.
You get a “standard” order relation by defining
$$
x\le y \quad\text{for}\quad x<y\text{ or }x=y
$$
Such a relation is indeed reflexive, antisymmetric and transitive and also a total order that satisfies


*

*if $x\le y$, then $x+z\le y+z$

*if $x\le y$ and $0\le z$, then $xz\le yz$.



In general, strict orders and orders are essentially the same thing. A strict order on $X$ is a relation $R$ that's irreflexive ($x\mathrel{R}x$ holds for no element) and transitive.
If $R$ is a strict order on $X$, then $S=R\cup\Delta_X$ is an order on $X$, where $\Delta_X=\{(x,x):x\in X\}$. Similarly, if $S$ is an order on $X$, then $R=S\setminus\Delta_X$ is a strict order.
In the case above, $\le$ is the order associated to the strict order $<$.
