cut out
1. 切去，割去，剪去；删去：
The doctor cut out his tonsils.
医生切除了他的扁桃腺。
You should cut out the unimportant details.
你应当删除掉这些不重要的细节。
2. 裁剪；剪出：
The shoemaker carefully examined the leather before starting to cut out the uppers.
鞋匠仔细察看了这张皮子以后才开始裁剪鞋帮儿。
3. [口语]击败；取代；排挤：
Mary was going to marry Charles but Jim cut him out.
玛丽本来打算和查尔斯结婚，但是吉姆把查尔斯排挤掉了。
4. [口语]戒除，停止服用：
I have decided to cut out smoking and drinking.
我已决定戒烟戒酒。
5. 停止；中断；关掉：
Let's cut out the talking and get back to work!
咱们别聊天了，回去干活吧!
6. (突然地、秘密地或永久地)离开：
A lot of kids cut out before the class was over.
还没有放学，很多孩子就悄悄离校了。
cut down
1. 削减
He tried to cut down on smoking but failed.
他试图少抽烟,但没成功。
2. 缩短
Cut down the article so as to make it fit the space available on the paper.
把文章缩短一些,这样就能排进报纸有限的版面中。
3. 砍倒
追问1，government spending that would make everyone better off was being cut down.
2, ……but only cut out jobs while keeping the money value of what those jobs produced for themselve
追答cut down应该是削减的意思吧
cut out应该是停止的意思吧
我也是猜测，仅供参考！