Rather than imitate Matt Emerton's multiple comment style, I'll just make my answer community-wiki (since there is much to say from different viewpoints). The first lesson is that you have to insist on definitions in each source you consult, since people sometimes use terms a bit differently. (As someone once said, the beginning of wisdom is total confusion.)
It's always important to define your context, as others have indicated. Historically, Lie groups (and Lie algebras) come first, initially over $\mathbb{R}$ and then in complex versions. There's no need to retrace all the history, since it took quite some time for the local theory of analytic and Lie groups to take global form and then be translated into Lie algebra language. But there is always a problem about dealing with non-connected Lie groups, which can even have infinitely many connected components. Here you just have to be cautious.
Initially Lie groups tended to divide into solvable and semisimple types, with
a Lie group called semisimple precisely when its solvable radical is trivial, or equivalently its Lie algebra is semisimple (has zero radical). The Lie algebra doesn't see connected components other than the one containing the identity, being essentially the tangent space to the group at that point. While the real case is most natural at first, the study of structure tends to translate mostly into the Lie algebra setting where complexified versions are much easier to start with. (Then you have to go back to real forms, etc.) But the algebraic theory in the semisimple case is fairly elementary though tricky and leads to a good classification.
In the Lie algebra setting, it's easiest to define a real or complex finite dimensional Lie algebra to be reductive if its solvable radical equals its center; this definition can then be used for connected Lie groups, though the disconnected case tends to get messy. The word "reductive" and the motivation for its use arise in group theory via the notion of complete reducibility of (finite dimensional) representations. But as you can see in the introductory pages of one of Knapp's books, it's tricky to apply the term successfully to Lie groups. In his 1965 book Structure of Lie Groups Hochschild defined a complex analytic group to be reductive just when it has a faithful finite dimensional analytic linear representation and moreover all such representations are semisimple (= completely reducible). That's the spirit in which Hochschild and Mostow studied the groups, characterizing the real linear ones in terms of symmetry under transpose; but here you need a fixed linear realization.
Anyway, it's usually best to proceed with an intrinsic characterization of the groups or Lie algebras. While most of the structure theory focuses on the simple or semisimple cases, modern work (especially involving Harish-Chandra's program and the Langlands program) works best with reductive groups by allowing induction from parabolic subgroups with reductive Levi factors to play a major role.
P.S. In the algebraic group setting, much but not all of the structure theory goes through in a similar way, but finite dimensional representations are usually not completely reducible; even so, "reductive" groups are commonly used along with "semisimple" groups and people get confused. In the algebraic version, reductive doesn't require connected but does require that the unipotent radical be trivial (while the solvable radical of an algebraic group is the semidirect product of a torus and a unipotent normal subgroup). In any case, radicals are required to be connected normal subgroups.
Surely it's easier to check whether $H'$ is simply-connected by inspecting the co-root lattice...? For the example of $G_2$ containing $SO(4)$ that Allen mentions, we have a pseudo-Levi subalgebra of type $A_1\times \tilde{A}_1$ where $\{(3\alpha+2\beta),\alpha\}$ is a basis of simple roots. Now the cocharacter $(3\alpha+2\beta)^\vee=\alpha^\vee+2\beta^\vee$, so the lattice:
${\mathbb Z}(3\alpha+2\beta)^\vee+{\mathbb Z}\alpha^\vee = {\mathbb Z}\alpha^\vee+{\mathbb Z}(2\beta^\vee)$
is of index two in the cocharacter lattice ${\mathbb Z}\alpha^\vee+{\mathbb Z}\beta^\vee$ for $T$.
In fact this allows you to determine exactly what the pseudo-Levi subgroup is in each case.
For the maximal pseudo-Levis there's an easier trick to find non-simply-connected ones: if $s\in T$ and $L=Z_G(s)$ then $Z(L)/Z(L)^\circ$ is generated by $s$, by a result of Eric Sommers. So we can see almost immediately that hardly any maximal pseudo-Levi subgroups are simply-connected. For example, the pseudo-Levi of $F_4$ which is of type $C_3\times A_1$ has a cyclic centre, so it can't be isomorphic to $Sp_6\times SL_2$. Specifically, it is isomorphic to $(Sp_6\times SL_2)/\{ \pm (I,I)\}$.
EDIT: A mistake with this is that Sommers' result only holds for adjoint type groups. More generally we have $Z(L)/(Z(L)^\circ Z(G))$ is generated by $s$. Of course this makes no difference for type $F_4$.
Best Answer
As indicated in the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis in the Chevalley seminar 1956-58 was on achieving a uniform classification of semisimple algebraic groups over an algebraically closed field of arbitrary characteristic. For this a slightly more general treatment of isogenies between such groups is more natural. (Chevalley discovered for example that the simple groups of types $B_\ell$ and $C_\ell$ are isogenous in characteristic 2 while their groups of rational points are isomorphic even though the underlying algebraic groups are not).
A little later, Demazure and Grothendieck translated most of this into the more flexible language of group schemes in SGA3, while Borel and Tits by 1965 expanded the framework by defining reductive algebraic groups over an arbitrary field. (For such groups Tits achieved a classification, modulo some later reformulation of his theorem stated in the proceedings of the 1965 Boulder AMS Institute. The main unsolved problem is to classify the $k$-anisotropic groups, which vary a lot for different fields of definition $k$.)
The Chevalley seminar and the other sources in French are available online from numdam, but note that SGA3 has been re-edited in recent years while a corrected typeset version of the Chevalley seminar was published in 2005 by Springer. (In my 1975 textbook, I mostly followed the Chevalley seminar; but when the characteristic is not 2 or 3, my 1966 thesis showed that it's also possible to rely more on the Lie algebra as in the classical characteristic 0 case.)
Variants were found by M. Takeuchi and T.A. Springer, using for example ideas of Serre and Steinberg about generators and relations for the groups. But in spite of the differences in the published approaches, all require a lot of detail about the internal structure of simple algebraic groups (those with no proper closed normal subgroups): Bruhat decomposition, generation by tori along with root subgroups. A key conclusion is that two such algebraic groups are isomorphic precisely when they have the same root system (or Dynkin diagram) and the same fundamental group, except in type $D_\ell$ with $\ell >2$ even, when you have to distinguish the half-spin and special orthogonal groups. (Note too that the work of Tits gave a more precise picture of the internal structure: when $G$ is a simple algebraic group, its only proper normal subgroups are those contained in the finite center.)
Using the methods of Chevalley (1955), one further shows the existence of all possible types of simple algebraic groups. The classification of possible semisimple groups is then a routine but slightly messy exercise: start with a product of simply connected simple groups, then factor out a subgroup of the (finite) center. There may be a great many possibilities.
Translating all of this into the language of reductive groups is then a matter of reading between the lines in Springer's textbook: given an isomorphism of root data, one gets an isomorphism of root systems along with a comparison of fundamental groups, etc. Unfortunately, there is no single source in the literature for a truly unified treatment of all these matters, including isogenies. But the core of it all is the study of the Borel/Chevalley structure theory. The transition to reductive groups is needed mainly because these are more natural for induction purposes than the semisimple ones: Levi subgroups of parabolics are reductive but seldom semisimple. However, central tori over an algebraically closed field are fairly innocuous.