A Lie group is semisimple if and only if its Lie algebra is semisimple which in turn is equivalent to saying that the Lie algebra is a direct sum of simple ideals.
This means that a Lie group is semisimple if and only if it is locally isomorphic to a direct product of simple Lie groups.
If the Lie group in question is connected and simply connected, it follows that it then is a direct product of simple groups.
A connected Lie group is simple if and only if its Lie algebra is simple and this is equivalent to saying that all its normal subgroups are discrete and one shows that every normal subgroup must be central then.
With your definition, this means that every almost simple group is indeed simple, but not the otehr way round, as the universal covering group of $SL_2({\mathbb R})$ shows.
This is not the way a mathematician wants it.

The notion of almost simple groups is not used in Lie theory, but in the theory of algebraic groups, see for instance the book of Margulis. Indeed, if your Lie group is a linear algebraic group, then its connected component is simple as a Lie group if and only if it is almost simple.