After review of logic, the count within the mini cart always performs "Display number of items in cart" regardless of Admin setting. As a note: this is for the count within the Mini cart.
I reproduced this issue on stock Magento 2.2.3 with Sample data, however, I forgot to take a screenshot. Below is a screenshot of a recent project displaying the same issue.

Hello @duffner, thank you for your report.
We've acknowledged the issue and added to our backlog.
Closing as not a bug. The settings Set Sales -> Checkout -> My Cart Link relates only to the number of items shown in the box near LOGIN link, i.e. the link which opens the cart
@slavvka, what you're saying is correct, however that's the bug. The feature is not consistent as there are two locations for item cart count in the "mini cart" once in the header display, the other within the minicart itself.
Please ask questions before "closing" if you do not understand.
@slavvka If it's not a bug, please answer the following question . "How can you have 5 items and 2 items in the cart at the same time?" This makes absolutely no sense.
@duffner thanks for catching that. In Luma theme we have "items" label hidden with CSS so didn't see the inconsistency.
The correct behavior would be:



@tkacheva, thanks for providing the additional information! I test off of base, not Luma, I guess I need to specify this in future tickets. Once again, thank you
Solution to problem described:
<span class="count" text="getCartLineItemsCount()"/> to <span class="count" text="getCartParam('summary_count')"/>@Volvoxpl I tried your recommendation, It says unable to process binding. getCartParam is not defined
Any thoughts?
I was editing cart-items.html for checkout page, not minicart. But to fellow explorers who might want to show total cart items in checkout area, replace getCartLineItemsCount with getItemsQty
The involved js file is vendor/magento/module-checkout/view/frontend/web/js/view/summary/cart-items.js
here are my notes https://tutes.in/how-to-show-correct-item-count-in-magento-2-checkout-order-summary/
The method mentioned by @Volvoxpl works fine for minicart because it involves js
vendor/magento/module-checkout/view/frontend/web/js/view/minicart.js
Most helpful comment
Please ask questions before "closing" if you do not understand.