Okay, that looks better 
I have looked at it myself on desktop and mobile and you get the slide page transition on desktop and only the show page transition on mobile, as you have described.
I have simply opened up a Corvid Example and changed the page transitions to horizontal to match your site and you still get the same effect happening.
It works on desktop and does not work on mobile.
Therefore, it will only work on desktop or any table device that shows the desktop layout.
On any mobile devices the horizontal page transition is not allowed and so not possible and therefore is changed by default to the simple fade in.
Finally, as the page transitions is not a code related issue as it is purely set in the Wix Editor page settings itself, there is no code workaround to fix this for you.
If you want more help with page transitions then you would be better suited going to Wix Support as this is not a code related issue and therefore not a Corvid Forum issue.
Sorry for not being able to help you anymore.
The only thing that you could try with code is to have all the elements on the home page which you move to after a set time, to have them all setup with the slide effect through Wix Effect Options and to be shown like this on mobile only devices.
https://www.wix.com/corvid/reference/$w.EffectOptions.html
https://www.wix.com/corvid/reference/wix-window.html#formFactor
Mobile only Wix tutorials.
https://support.wix.com/en/article/corvid-tutorial-displaying-elements-in-mobile-only
https://support.wix.com/en/article/corvid-writing-code-that-only-runs-on-mobile-devices
You can even set this mobile home page to be shown only on mobile too.
https://support.wix.com/en/article/showing-a-page-on-your-mobile-site-only
It won’t be the page transition that is sliding, but at least the mobile version might work with it set as this.
https://www.wix.com/corvid/reference/$w.EffectOptions.html#SlideEffectOptions
Examples
Show an element using the slide effect and options
let slideOptions = {
"duration": 2000,
"delay": 1000,
"direction": "right"
};
$w("#myElement").show("slide", slideOptions);