Just splitting a video and/or audio event doesn't create two separate events, so renaming one of the split segments will also rename the other segment/s because it's still the one event. To have segments of the one event with different names, you'd need to render each segment individually - by doing so, you'd be creating two or more separate events each able to be individually named.
I am away from my computer that has Vegas installed at the moment, but would saving the split events as a subclip (which I believe can be renamed) solve your problem? I will check when I get back to the office.
You can create subclips either in the trimmer or from the timeline without rendering them, and give them different names. They will appear as new clips in the project media window. From the timeline, select the event you want to convert to a subclip, right click with the mouse and choose "create subclip" from the menu.
Having created your subclips the easiest way to rename them is to go to your Project Media window where they should have appeared. Right click on each and rename.
You can create subclips either in the trimmer or from the timeline without rendering them, and give them different names. They will appear as new clips in the project media window. From the timeline, select the event you want to convert to a subclip, right click with the mouse and choose "create subclip" from the menu.