When you have 2 or more layouts of different dimensions (not just swapped dimensions, in which case the layout changes with orientation), it's the resolution of the device/browser on which the content is being viewed that determines which layout is used (as opposed to the naming of the layout). Here's a video that shows setting up alternate layouts for orientation and for browser width:Creating Responsive Liquid Layouts with InDesign & in5
What is the resolution of the phone where you want the mobile layout to be displayed, and what are the dimensions of the other layouts? It sounds like the dimensions of the layouts may need to be adjusted.
It's still the width, not the name, that matters.
If your alternate layout dimensions are as follows:
then you should see the mobile layout for iPhone 5s (with a width of 320 px) but the Custom V layout for most iPhone 6-8s (with a width of 375). The Custom V layout will be displayed until the width of the browser is 360 px or less.
Here's more information on pixel dimensions:How do I know which pixel dimensions to use for my mobile layouts?
If you're still having trouble with the mobile layout showing for the iPhone 5 version, would you be able to send us a zip file containing your InDesign file and your html5_output? You can send it to: https://ajarproductions.com/pages/contact.php?category=in5