This has ALOT to do with IFS therapy and little to do with DID (especially if your T said you are not DID).

IFS believes that everyone has parts, that we are all made up of various parts that react and behave according to where they are. So yes, everyone has child parts that are stuck in the behaviors or actions or ways of thinking that a child is at the time of trauma or at the time that was very upsetting. There can be child parts that exist at different ages and stages. It is a way of saying, you have behaviors that are childlike and regressed. So, for example, you might have a part that soothes itself in a harmful way. That part is stuck in a time where it needs to learn to be soothed or soothe itself in a healthy way. Unburdening it is to find out what the part needs, helping it gets it need met, and allowing the Self to understand it.

IFS does believe in DID, but the difference between DID and "normal parts" are that DID parts are more extreme in their behaviors and are separated usually by amnesia. So the Self is not aware of the parts. (this begins to get more like DID therapy)

QUOTE from you:
"My t has several times said things like, "We need to help the small parts become unburdened when they are ready to do so" and "I want to invite these parts to come into the present when they feel it is safe enough."

The above quote is typical IFS language. The bells are ringing!

I suggest that you look up Richard Schwartz who created IFS and get his book. It will tell you everything you need to know. It won't tell you how to do the therapy, but it will help explain what your T is doing and what the parts are that each of us has inside us. And the concepts of unburdening the parts to free the Self.