Is the client's 'Engage date' the same as when their Letter of Engagement was signed?

No. The Engage date defaults to the date the client record was created in Engager, and you can change it in Step 1: General info. Signing a Letter of Engagement doesn't change it.

To see when the Letter of Engagement was actually signed, check the client record's Proposal, LoE & LoD section, under Documents. The signing date is on the letter's Decision line (for example, "Decision: Accepted" and the date), with Signed by showing who signed it.