Part of what makes this question a bit thorny is the Western position that Russia already had troops in the breakaway regions. So arguably Russia sending in troops could be considered them bolstering that force rather than a proper invasion. The Biden White House itself at first avoided, in the words of Politico, "the 'i' word," and The Washington Post reported that the White House was "wrestling" over whether to call it an invasion and even suggesting to reporters that it might not be.
Still, the administration official repeatedly refused to say whether
Putin’s decision to send “peacekeeping” troops into the two
Russian-backed separatist areas constituted a red-line invasion in the
eyes of the Biden administration. If anything, the official tried to
portray Monday’s developments as far short of a dramatic change in the
“Russia has occupied these regions since 2014,” said the official, a
point he emphasized several times throughout the call. “It has been
Russia’s position that there are not Russian forces present in this
part of the Donbas. The reality, as we pointed out on a number of
occasions over these past years, has been quite different. There have
been Russian forces present in these areas throughout.”
After the call, a different administration official defined a Russian
invasion that would prompt a clear U.S. response as crossing into
Ukrainian territory that Russia has “not occupied since 2014.”
This would make sense on its face. When the United States sent a surge of troops to Iraq in 2007, was that an "invasion"? Common usage of the word would suggest, no, the term is reserved for the initial deployment of forces in 2003.
But as of this morning (February 22, 2022), the Biden White House has begun to use the term "invasion." An official told the AP that "The White House decided to begin referring to Russia’s actions as an 'invasion' because of the situation on the ground... The administration resisted initially calling the deployment of troops because the White House wanted to see what Russia was actually going to do. After assessing Russian troop movements, it became clear it was a new invasion, the official added." But mostly this was done for practical reasons: the West set "invasion" as a red line that would trigger certain sanctions, and the Russian aggression is bad enough that they would like to put those sanctions in place. The Biden White House is not terribly invested in whether Russia's actions fit the dictionary definition of "invasion."
It's worth noting that the Western internal debate is not actually that deferential to the Russian position, since if these actions aren't an "invasion", that's only because the "invasion" was in 2014. Either way, a whole bunch of Ukrainian territory is transferring to Russian control, and the Russian Army is doing it.