For starters, I don't leave much spare wood at the base. For me, all the extra wood you show at the bottom of the vessel in the photo would be in my way. And instead of taking what was the tenon down to a dowel and sawing it off, I take it down to a cone and turn it off. These pics are from a larger bowl, but I do the same thing on most smaller pieces.
First, I reverse the piece into whatever kind of jam or friction chuck I have handy. For this bowl, I used another rough-turned bowl that was a bit smaller than the mesquite bowl I was making. I mounted the rough bowl onto the chuck and trued it up first.
Next I used some neoprene shelf lining pad between the two bowls, and mated them face to face (one inside the other), using the tailstock to hold them together.
I made sure the live center went into the original divot that was made with the tenon was first turned.
Then I gradually turned the tenon down into a peg that's about the size of the cup on my live center.
At this point on this bowl, I added some plastic stretch wrap as a "safety net" in case something went wrong with the rest of the operation. It's not mandatory, but prevents the accidental launching of the nearly finished bowl across the shop.
Then I started turning the peg into a cone...
My 3/8" shallow detail gouge is ideal for this...
Taking very light cuts at the end, eventually the cone is completely severed from the bottom of the bowl. In this pic, the cone is not spinning, but you can see the bowl still is.
Since I had stretch wrap on this piece, I was able to move the tailstock away a little bit and make a few very light finishing cuts on the bottom. Without the stretch wrap, I'd have done the final smoothing off the lathe with sandpaper, either with my 2" pneumatic ROS, or by hand...
And a shot of the finished bottom.
For hollow vessels and vases, the process is pretty much the same as I did here, although the scrap wood friction chuck might be on the outside of the project piece instead of on the inside like this one was. It just depends on the size and shape of the project piece, and what rough-turned scrap pieces I have laying around the shop. I have a vacuum chuck, but I've found the jam chuck is as effective, and generally faster and easier for me to set up.
Also, with the exception of smaller low-priced pieces like potpourri bowls or Christmas tree ornaments, I prefer to apply the finish off the lathe, after the bottom has been finished. That way I don't have to worry about damaging the finish with whatever jam chuck I end up using, including the vacuum chuck. (Plus, 99 out of 100 times, I can spot concentric rings in the finish of pieces that are finished while spinning.)