Do I have to click on "save for later" and create a wish list, which I know I can share, or is there a more direct way to share my shopping cart?
You can't directly share your shopping cart directly, I mean you can't share links of your cart by anyway because of security.
So most usual and used way to do that is by sharing your 'wish list' or you can use an extension like
Shopping Cart Share For Amazon
Had the same issue recently and didn't find a solution. So, I made a cart sharing extension for Chrome browser. Here you go: https://chrome.google.com/webstore/detail/amazon-share-a-cart/peiobddfomnijgoblelfdmjblplojcoe