First of all, Dirichlet series means any series of the form $\sum_{n=1}^{\infty} a_n n^{-s}$, while Dirichlet $L$-series means something much more specific: a particular Dirichlet series with $a_n = \chi(n)$ for $\chi$ a Dirichlet character of some conductor.
The terminology becomes even more confusing than that(!), because there are more general $L$-series (e.g. those attached to elliptic curves over $\mathbb Q$, which
are the subject of the BSD conjecture), which are Dirichlet series in the general sense, but are certainly not Dirichlet $L$-series.
So the three classes of functions, from most general to most special, are:
$$\text{Dirichlet series} \supset \text{$L$-series} \supset \text{Dirichlet $L$-series}.$$
(Note that people often say $L$-function rather than $L$-series; the two expressions are synonymous.)
There is not much that one can say in general about Dirichlet series; if you choose the coefficients $a_n$ at random, you will get a pretty badly behaved function. (Even if it converges on some right half-plane, it typically will not admit analytic continuation to the whole complex plane, will not have a functional equation or satisfy any sort of Riemann hypothesis, etc.)
Now $L$-functions are particular Dirichlet series arising from the theory of automorphic forms and the theory of Diophantine equations. They are conjectured to be very well-behaved, although this is not known in general. In particular, each $L$-function $L(s)$ has a conjugate $L$-function $L^*(s)$, and is conjectured to satisfy a functional equation
relating $L(s)$ to $L^*(k-s)$ for some integer $k$.
(In lots of interesting examples one has $L(s) = L^*(s)$; this includes the $L$-functions attached to elliptic curves over $\mathbb Q$, Dirichlet $L$-functions for quadratic Dirichlet characters, and $\zeta$-functions of number fields such as Riemann's $\zeta$-function (which can also be thought of as Dirichlet $L$-series for the trivial character).)
One general form of the Riemann hypothesis than says that all the non-trivial zeroes of
$L$ lie along the axis of symmetry for the functional equation, i.e. along the line
$\Re s = k/2$.
Note that for Dirichlet $L$-series, the integer $k = 1$, and so this gives the Riemann hypothesis as it is usually stated, namely that the non-trivial zeroes lie along the line $\Re s = 1/2$.
For the $L$-series attached to an elliptic curve, one has $k = 2$, and so the non-trivial zeroes should lie along the line $\Re s = 1$. In particular,
$s = 1$ is allowed, as in the BSD conjecture.
Now if we are given an $L$-series, note that we can make a substitution of $s+c$ for some $c$ into $L(s)$, and also into $L^*(s)$,
to get new functions $L_1(s)$ and $L_1^*(s)$, which will satisfy a functional
equation relating $L_1(s)$ to $L_1^*(k - 2c - s).$ Thus if we choose $c = (k-1)/2$, we will have a functional equation relating $L_1(s)$ to $L_1^*(1-s)$.
This is called the unitary normalization of $L$-functions, and as it is a very simple change of variables, it is not difficult to translate properties (known or conjectured) of $L$ to those of $L_1$. For $L$-functions with the unitary normalization, the Riemann hypothesis is the statement that all non-trivial zeroes satisfy $\Re s = 1/2$.
Analytic number theorists often consider just unitarily normalized $L$-functions; for people working with Diophantine equations, non-normalized $L$-functions are often more natural. So, for example, the BSD conjecture is always stated in the non-normalized form, with regard to the point $s = 1$. If we rewrote the BSD conjecture for the unitarily normalized $L$-function (as we could), it would refer to the value of the unitarily normalized $L$-function
at $s = 1/2$.
As Matt Young explained in the comment thread to your post on MO, the convergence of the infinite product, or of the Dirichlet series, is not known
at $s = 1$. They certainly do not converge absolutely there, but they are conjectured to converge conditionally (at least, the Dirichlet series is),
but this is not known, since to get this conditional convergence one would need
cancellation between the different terms in the series, and this degree of cancellation has not been proved.
On the other hand, the $L$-series for an elliptic curve over $\mathbb Q$ is known to analytically continue over the whole complex plane; this was proved by Andrew Wiles, Richard Taylor, Christophe Breuil, Brian Conrad, and Fred Diamond.
(So the question of whether the series converges at $1$ is the question of whether a particular representation of the analytically continued function is possible, not whether the analytically continued function exists --- the latter is not in doubt.)
If you are interested in general properties of $L$-series, you may want to read about the Selberg class; this is Selberg's conjectural characterization of the most general class of $L$-series satisfying a Riemann hypothesis. Note that he (implicitly) assumes that the $L$-series are unitarily normalized (see conditions (2), (3), and (4) of the linked article), so his Riemann hypothesis is stated in terms of $\Re s = 1/2$.
There are also other questions on Math.SE about zeta-functions and $L$-functions and the Selberg class, which are also related to the issue of normalizations that concerns you, and which you may want to look at, e.g. here, here, and here.
