TITLE: Definitions of Reductive and Semisimple Groups
QUESTION [12 upvotes]: I'm a graduate student.  I've been reading Knapp's two books Representation Theory of Semisimple Groups and Lie Groups Beyond an Introduction.  He seems to give wildly different definitions for the basic terms, 'reductive' and 'semisimple', and it leaves me unclear on what to think when I read 'reductive group' or 'semisimple group' alone in the literature.
To be more specific, Knapp doesn't actually define "reductive" in Representation Theory; he defines "linear connected reductive group".  But in his other book, one of the reasons he says we need to introduce reductive groups is to deal with the disconnected groups arising from our analysis.  Their disconnectedness is "not too wild", and so much of the structure theory developed for semisimple groups goes through.  The definitions in Lie Groups make more sense to me.  
I'm similarly confused about whether semisimple groups have finite center, because Knapp seems to disagree with himself on that.  
How should I think about what these terms mean?  Is there an authoritative definition.  The definitions in Representation Theory seem most strange to me, but the resulting theory he develops is very familiar from everything else I've read.  I'm curious why Knapp chose such strange definitions for his Representation Theory in the first place.  Does the substance of his book go through if I replace his definitions with something more standard.  (I'm aware this last bit is rather vague.)

REPLY [4 votes]: In addition to other answers, especially if one is interested in (infinite-dimensional) representation theory of Lie groups, a common technical issue is that the maximal compact subgroup should meet all the connected components of the ambient groups. This is part of what often gets called "the Harish-Chandra class"' definition for semi-simple or reductive, and is (apparently!) often sufficient to cope with disconnectedness of the ambient non-compact group.
And, reiterating a point made in the other answers, one always must pay attention to context to know the precise assumptions, e.g., about connectedness and/or what compromises suffice for the mechanisms to succeed without assuming the ambient group is connected.