There is a related open issue here:
The problem basically comes down to the fact that Font Awesome (and presumably ionicons, from what you've described) have changed their naming conventions, but which convention you use depends on what version of FA or ionicons you're using.
So, for FontAwesome, you can see the css that's imported for R Markdown here:
(Or as webfonts here: https://github.com/rstudio/rmarkdown/tree/f111cf9601393317bac9cbc0e6dc81336c4fe002/inst/rmd/h/fontawesome/webfonts )
Ionicons looks like it's at v 2.0
There is an icons package that's dedicated to icon rendering, but IIRC (or, last time I ran into this), these weren't necessarily all updated either: