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Little confused about what you're after so should clear that up first.
So I read this as a yes, you're trying to do what I asked.
It's a mathematical thing, with what you're doing you're creating an odd amount of vertices, which makes it impossible to not have a triangle. So basically, you have to add enough vertices in such a way that you have enough geometry left to create quads.
But that can possibly create other issues for you too, down the line. Depending on how much detail you want to add, managing the topology as you increase it could end up getting difficult to manage.
Let me grab you a quick screenshot here of how I increase the amount of faces similar to what you're trying to do but without a triangle.
So I have a row with 5 faces, to increase that to 6 faces without the triangle you're getting, I add another edge loop (the purple line is the new edge loop) that comes in from the side. And now my next row has 6 faces, but the mesh is still all quads.
You'll see that there is a 5 sided pole there now, but to do what you're asking you will have to put up with poles or triangles at somepoint.
And obviously, when you add an edge loop like this, now you've more topology you have to deal with running in another direction. Not necessarily an issue depending on your goals. But it can make things tricky.
https://www.youtube.com/watch?v=llOqbJKpK1Y&list=PL3GeP3YLZn5irhqsD6_Srf-CeimYPonaK&index=7
https://topologyguides.com/
EDIT: Not the video I thought it was, I might not have bookmarked the one I'm thinking of, I'm still seeing if I can find another topology management video for you.
https://www.youtube.com/watch?v=HGL6QpVRyXk
And a little pointless, you don't save much but get a lot of issues. In the case of your vase, you would simply use the same poygon count around the thinner and the thicker part.
If you really need to transition from fewer to higher poly numbers (or vice versa), you can use this handy chart:
https://topologyguides.com/assets/img/163679954765_0.png
Thanks, still__alive, for the link, would have had to search for it otherwise. ;)