Yes, both plants are the
Dragon Tree (Dracaena reflexa var. angustifolia) It was formerly classified as Dracaena marginata and I still call it by that name because I'm old and that's what I've known it as for 50+ years.
The stems on the plant that you've cut back appear to be dry and desiccated; I'd cut those stems back even more, a little at a time, to see if there's any viable green wood remaining. What type of soil are you using? To my eyes, it looks very dense and dry, which may be a problem.
I just went outside and cut a stem off of one of my dragon tree plants to show what you should be seeing on a viable stem. I scratched away some of the tan/brown bark so you can see the green, viable wood peeking out beneath.
The main plant where I lopped off the stem.
This is the top part of the stem that I removed and I'll stick it in soil to root a new plant.
I have a few Dragon Trees growing in my garden and also a few that I keep as container plants that stay out on my screened porch year round. I find them to be fairly drought tolerant and depending on the type of potting medium/soil being used and how quickly it dries at root level should determine how often you will need to water. You should try to keep the soil barely moist, not overly wet and not extremely dry for too long. If the soil at root level remains too wet, the roots will suffocate and die. I always err on the dry side with my plants.
