Attaching external mic to MacBook Air Mid 2011
The splitter in your photo has a TRRS jack, which is often called "4-pole" or "4 position" because of its 4 metal sections separated by 3 bands.  

TRRS enables both stereo output as well as microphone input.  (Earbuds from iPhone 6 use this type of jack.)

Your MacBook Air (although I'm not familiar with yours in particular) probably has a TRRS port (rather than one port for microphone and a separate port for headphones).  So you could plug iPhone earbuds (which have a mic) into that port.

The lavalier mic shown in your photo appears to have a TRRS jack and so would probably work with your MacBook Air without even using an adapter.

However, you said that you own a different one and that yours has only "one ring". 

I would agree with your assumption that plugging a 1-pole mic jack into that adapter and plugging the adapter into your MacBook Air should work.

Many people online are reporting problems with this MacBook Air port, however. 

See my unanswered question here: External mic not working anymore on Macbook Air

By clicking the links mentioned in my question and comments, you might stumble onto some hacks relating to:


Putting your computer to sleep after plugging your mic in
The order in which you plug your mic into your adapter and your adapter into your laptop
Whether and when to plug headphones into the adapter
and more


I have not yet found a reliable repeatable solution but wanted to share what I've come across so far, and I hope it helps you (and others).