It can get a bit confusing at times, the carbide insert/bit is put into a tool holder, the tool holder is put into a tool holder, then put on to the post.
I dislike ambiguity too. In the old days, it was simpler. You had a tool post and a tool. The post was usually a lantern style. For me at least, clarity is achieved by adding words when they are needed. If I think my intent is obvious or doesn't matter, I'll often drop the extra words.
For example, on my current lathe, I have a wedge style tool post, then tool holders, then insert tool holders, then anvils, and then carbide inserts. Piston style tool posts, and 4-way tool posts are both popular too. I usually add "carbide" before inserts because there are also HSS inserts out there.
I know that different manufacturers use different names for some of these things, and I also know that other people use different names too. I'm not claiming that my way is the right way. I only suggest it because I believe that it works for me - most of the time.
The one that causes the most confusion is that anvil. Lots of people ask me what that is when I mention them because lots of people, and also many manufacturers, prefer to call them shims. I wish they wouldn't, but I can't change the world.
A few of my other pet pieves are indicators, test indicators, and dial test indicators..... And let's not forget Vernier calipers, dial calipers, and digital calipers.
So much confusion out there! LOL!