We are experiencing similar issues, too. Slicers are working fine in Power BI Desktop, but then in Power BI Online, they break as described by the OP. Mainly on one report that has a lot of cross-filtering and case statements to make things work. And uses the Chiclet Slicer.
Just to be sure, I changed one other report to use the Chiclet Slicer, and it behaves fine. So it appears more complex filtering (in my case, at least) may contribute to the issue.
I know it's very strange, but I don't see these problems with slicers when I publish my report to My Workspace. Whereas the same report has these problems when I publish to other workspaces. Also, in my case, I don't see the problems with the standard, OOB slicer. I'm getting the problems with Attribute Slicer and specifically when I publish the report with some items selected and others unselected.
We'll now I see the problem in My Workspace with another report. So problems occur in My Workspace also. It's really hard to pin down when this problem occurs and when it doesn't. It's very baffling. And very annoying. I wish Microsoft would investigate and fix it.
I'm having a similar issue but it appears on MS Edge and not on Chrome... I had a client describing strange slicer behavior, he showed me via Skype and then I had him try Chrome. All worked as it should. I then tried Edge and had the issue.
I have some slicers with a slider on Desktop, and these work fine there. Online however, these change to a vertical ticklist. This takes up a different spacing than the Slider, which is more of a horizontal box.
It seems strange that this is a standard visualization on the desktop, but is not fully available in the online version.
I was trying out the new synch of sliders option. Works great in destop and online reports, but is not active in the online dashboard. Wonder if this option will come in later.