does lithification include diagenesis in a broader term?

Answer 1

Any process that affects a sediment AT OR NEAR earth's surface is diagenesis (excepting the movement of earth and sub-aerial erosion). Hence diagenesis is minor change in the sediments under low pressure and low temperature. It involves several processes like Pre-burial (disruption of sedimentary structures, mixing by action of organisms, changes due to Eh and pH conditions of the basin of deposition, development of certain authigenic material like glauconite, phosphorite etc.), Early burial (adjustments of Clay minerals to the chemistry of the basin water, development of more authigenic minerals, continuation of compaction, i.e., expulsion of entrapped water, primary cementation etc.) and Late burial (completion of dewatering of clays, completion of cementation, development of concretions, replacement of aragonite by calcite etc)
Lithification, on the other hand, is the process by which loose sediments change to massive rocks. It involves mainly two of the several diagenetic processes - compaction and cementation.
So you see, diagenesis is several processes that affect a sediment, but only a few of those processes actually control lithification.
Therefore, you can safely say that diagensis includes all the processes of lithification (but lithification does not include all the processes of diagensis, as you have mentioned)

Answer 2

Other way around. Diagenesis includes lithification. Diagenesis is the process of rock forming and lithification is in the last stages.