An option to tick (or default option in the admin section) to setup link to allow the direct opening with office desktop when you share a document could be interesting or to use the setting embeded in the site library from where it come (Open Documents in Client Applications by Default in the site collection feature)
Currenctly, if the site collection feature "Open Documents in Client Applications by Default" is activated, the sharing option is not the same and it confuse some users. If they go on the website, it open with word desktop, if they use the sharing like, it open word online..