Tulip is the common name.
greigii is a species name as is batalinii and I think the x greigii means a hybrid cross of that species. Many places, as you see, list the species a bit differently.
The problem I am seeing in many of the genus listings here at ATP is that some entries have just the cultivar name and some have the species name and that same cultivar name so now there are two entries for the same plant. When you add a species name, it drops the plant toward the bottom of the alphabetical listing so often someone comes in to search in a particular letter, say for instance the letter R for 'Red Riding Hood. They go down the list, don't see it where they expect to see it in alphabetical order so they make a new entry. Had they gone further down the list, past the letter Z where the species are now alphabetically listed, they would have seen the entry the wanted listed as Tulip (Tulipa greigii
'Red Riding Hood'. Personally, I like the list cleaner with simply the cultivar names but I do think eventually Dave will iron out the cultivar duplicates somehow. I also personally do my searches now by cultivar name unless of course they are a species.