There's no evidence that "today" is a PIE term, and plenty of evidence that it isn't. For example:
- Different IE languages have contracted different roots to make their "today" words. That's already enough to rule out common descent.
- In fact, the roots often don't even all have the same semantics: about half of them are "this+day" like German, about half are "preposition+day" like English, and maybe a few oddballs here and there. The ones that use prepositions often use different ones, like English "to+day" vs. Dutch "from+day" vs. Danish "in/on+day".
- And these differences are clearly not at all distributed based on descent; Sanskrit and German are certainly not more closely related than German and Low German.
- These contractions appear long after the languages had diverged, which again is usually enough on its own to rule out common descent.2, 3
- Many non-IE languages have constructions that are semantically just as close as to "today" and "heute" as they are to each other, like Japanese "honjitsu" and Arabic al-yawma.
So, if they're independent coinages, why do some of them look so similar?
Well, it looks (see Arabic, Japanese, etc.) like forming "today" as a contraction is the kind of obvious thing that many languages will hit on once the host culture reaches a point where it needs a word like that.
There are only so many things you can plausibly contract to get "today": "to+day", "on+day", "this+day"… And if two different IE languages both contract their words for "this+day", those words are often going to be cognates, so the result will often look similar.
But again, notice that many of them are not cognates, and it seems to be almost random which ones are. Which is exactly what you'd expect from different IE languages developing their words independently, but out of a small and mostly-shared pool of inputs.
So, what do etymologists mean when they say OHG "hiutu" is a contraction of "hiu+tagu" possibly influenced by Latin hodie?
First, you asked in the comments why Latin would affect German in the first place. That's easy. When people were speaking OHG, Latin was the language of their religion, diplomacy, civil administration, and (if you include vulgar Latin) much of their trade. Almost any OHG speaker who was literate was literate in Latin. And Latin influence on Germanic languages even goes back long before that—the Romans were hiring Germans as mercenaries, resettling them all over Europe, and converting them to Christianity centuries before Charlemagne.
Also, notice that some of the Germanic languages ended up with a direct calque of the Latin contraction, but others are "phonocalques"—contractions that stretch the Latin semantics a bit further, but sounds a lot more like the Latin, especially with "heute". That could be a coincidence, but it certainly makes influence look even more plausible. I don't know exactly how compelling that argument is, but remember, the etymological dictionaries only say "possibly influenced".
1. Some as late as modern times—Norwegian rejected "idag" as an invalid new coinage in the 1938 reforms. Obviously the Latin and Greek aren't that new, but they appeared well after the start of historical writing, which is still millenia after the breakup of PIE.
2. Also, some languages coined the same idea multiple times. Low German "vandage" ("from+day") replaced "hiudu" ("this+day") in historical times. (Earlier, "hiudu" apparently replaced "hi(u)digu", which is formed from the same roots, but contracted differently. But it seems plausible to me that "hiudu" could be a borrowing from OHG "hiutu" rather than a re-coinage, so let's ignore it.)