Roamio supports a very fast TE 3 interface with all HD Menus/Submenus.
Premiere supports a pretty decent HD main menu with PIP/TV window in the Main Menu or "My Shows" list. (can show/hide pressing Slow button, or Settings)
[Premiere] The Guide in also HD but has a slight more of a delay in scrolling (PgUp/Down) compared to Guide on a Roamio.
[Premiere] Seeking/FF/RW/Play/Pause in video runs just as fine with HD Menus enabled.
[Premiere] Certain submenus do not exist in HD (i.e. Settings, System Information Menu, Messages, Parental Controls) and will revert temporarily back to SD menus while in these menus thus removing the TV/PIP window while in these specific submenus.
Premiere, unlike Roamio, has the option to still switch to 100% old school SD menus regardless of what type of TV is connected. The older menus do run faster, but the SD menus lack newer features such as no PIP/TV window while in menu, no grouping in recordings (My Shows) and missing other familiar TE3 layouts. The Guide in SD also looks like a cartoon
TiVO Mini TE3, whether connected to a Premiere or Roamio, does not have any PIP/video window while going through any of its menus other than Guide. TiVo Mini is only capable of showing TV/PIP window in-menu if upgraded to TE4 (Hydro) which requires the host Roamio or Bolt (not compatible with Premiere) on TE4.