What is Detach Request in LTE?

It is the EPS NAS message that starts release of EPS registration context, either from the UE side or from the network side.

Why does Detach Request have dual significance?

Because TS 24.301 defines both a UE-originated Detach Request and a network-initiated Detach Request.

What should I inspect first in Detach Request?

Start with who sent the message, then read Detach type, then any identity or EMM cause information.

What usually comes after Detach Request?

That depends on the variant: normal detach often continues with Detach Accept, while switch-off detach and some network-forced cases move directly into local release behavior.