Interesting, I hadn't appreciated that Matrix was switching from fully open source to an open core model.
Matrix the protocol is fully open source. Element the matrix client is fully open source. Synapse the matrix server is fully open source. Dendrite, the next generation matrix server, is fully open source.
All of these are developed by the New Vector team. The only thing here that isn't open source is an external bridge to microsoft teams, a platform that itself is already closed source.
In this case, "open core" would probably look like the Matrix team closing the Element client and Dendrite homeserver, and then extending them with closed matrix event types.
Honestly, they could probably pull it off, but I doubt they want to go that route. They've been dedicated to federation since the beginning.
But "Element Matrix Services" (aka New Vector) launching and marketing a product/feature that is not open source is AFAIK something new, and it'd be very surprising if this is the last such feature they announce.
(Before today, they were one of the very few examples I knew of a VC-funded open source company where their product remained entirely open source.)
First of all, Matrix itself is a non-profit foundation, and absolutely everything it does is FOSS, and accepts contributions from everywhere. The majority come from Element (formerly New Vector) though, the for-profit that the original Matrix team set up in order to work on Matrix full time.
Element has always had some closed-source bits in addition to all the FOSS work we do for Matrix. For instance, the EMS hosting platform itself (lots of stripe + kubernetes) is all proprietary and always has been. The default integration manager (codenamed Scalar) is too. And occasionally we've built random apps and bots and bridges which we chose to keep closed source - e.g. we've had a closed-source Sametime bridge for ages, and we have closed-source border gateways, cross-domain controllers, audit bots, etc too. However, *this doesn't make Matrix open core*, any more than the existence of closed-source webapps makes the Web "open core". It also doesn't make Element "open core", any more than (say) Mozilla is "open core" just because they haven't released the source for some of the random services they run too.
In an ideal world, Element would release all of its IP as FOSS - and it's quite possible that we'll do that in future. But right now we're still experimenting with the right balance.
Matrix was never a great name, and confuses people. Riot was named after violence, but Element sounds like nothing, and now you've introduced a namespace collision for no good reason between the renamed product and the company.
What's next, rename Synapse to Element Server and Matrix to The Element Protocol?
It's like you want to lose
I would call that open core.