I import it with a USB connection, but My Photo Stream shows up without connecting my phone to my laptop with a USB.
That is abad mixture, if you are taking Live Photos. The problem is, that My Photo stream will only transfer the still frames of the Live Photos, and when you later try to import them by USB, they will be broken, because the video component cannot be paired with the still fame already in the library.
We still are in the dark about the system versions on your iPhone and Mac. Are you running iOS 11 on your iPhone? Then your iPhone may now be using a new video format and your Mac needs to be upgraded to High Sierra to be using this format, see this document how to adjust the settings on the iPhone and which system versions you need: Using HEIF or HEVC media on Apple devices - Apple Support