I agree with the Nova suggestion. I also make a variety of inserts. They are #2MT.
live_center_MT2_IMG_7914.jpg
This is what it comes with:
Nova_live_center.jpg
But to address your question, Izzy, I personally don't like the conical pointed live centers for some things since with a bit too much pressure they can split the wood, especially when turning a smaller diameter spindle. Not usually a problem with face work.
The Oneway and clone centers use a cup with a point in the middle - this is less likely to split the wood and the point can even be removed for a gentler holding. The points have a #0MT and the socket can also hold shop-made adapters, as can the threads:
live_center_MT0_C_IMG_7913.jpg live_center_threaded_IMG_7917.jpg
However, my favorite live and drive centers for spindles are Steb centers. They also have a central point but it is spring loaded (a huge plus in my opinion) so they can't exert too much force. The "cup" is a series of points which grip well in a variety of situations. I keep live and drive Steb centers in two sizes.
Steb_Sorby.jpg
JKJ