No, that's not what fraud is. Let me focus on the US legal system (different jurisdictions are different, though I suspect other jurisdictions are similar).

Fraud is the intentional deception to secure unfair or unlawful gain, or to deprive a victim of a legal right. The government does NOT try to officially define every phrase in the language, nor does it need to. That's not how it works. The issues are, (1) does there appear to be intentional deception, and (2) Would that deception result in unfair/unlawful gain or the loss of the legal right? Issue 2 is true by definition (open source software as the term is generally used always allows people to use the software for any purpose, including commercial users, and this doesn't provide that right, so by definition the victim is losing a legal right). So we're really only asking the first issue - is there intentional deception? I presume you'd argue that there's no deception. Only a court can decide that for sure, but if someone uses a phrase likely to mislead most people, that's at least getting dangerously close.

So no, a government doesn't have to define the term. But even if you think that governments have to use the term that way... well, governments do define the term "open source software" just like OSI does. Again, I'll focus on the US, but this is by no means limited to the US.

The US Office of Management and Budget (OMB) memo of August 8, 2016 (M-16-21) "Federal Source Code Policy: Achieving Efficiency, Transparency, and Innovation through Reusable and Open Source Software" https://www.whitehouse.gov/sites/whitehouse.gov/files/omb/me... defines the term "open source software" as follows:

"Open Source Software (OSS): Software that can be accessed, used, modified, and shared by anyone. OSS is often distributed under licenses that comply with the definition of “Open Source” provided by the Open Source Initiative (https://opensource.org/osd) and/or that meet the definition of “Free Software” provided by the Free Software Foundation (https://www.gnu.org/philosophy/free-sw.html)" Notice that it can be "used by anyone" and it specifically references the OSI's definition. That memo included a mandate to release a certain amount of code as OSS - which meant the government had to define the term, and yes, they used the normal definition for it.

That OMB memo cites the US Department of Defense (DoD)'s official policy on open source software that was released in 2009: https://dodcio.defense.gov/Portals/0/Documents/FOSS/2009OSS.... - it says in 2.2.b.1.iv "Open source licenses do not restrict who can use the software or the fields of endeavor in which the software can be used." So the US DoD thinks that commercial use is by definition allowed by open source licenses. This wasn't even the first government memo about open source software; they had another one in 2003.

I've been citing executive branch policies, but it's also in US law. The National Defense Authorization Act (NDAA) of 2018 at https://www.congress.gov/bill/115th-congress/house-bill/2810... section 875 requires the DoD to participate more fully in a "pilot program for open source software", and cites the policy and definition of OMB M-16-21 ("Federal Source Code Policy: Achieving Efficiency, Transparency, and Innovation through Reusable and Open Source Software") dated August 8, 2016. Yes, that's a US law, and it by reference defines "open source software" using the OSI definition. That is, commercial use must be allowed.

So yes, the US government DOES have a definition of open source software, and it requires permission for any field of endeavor, just like OSI's definition does. It's been that way for almost 20 years.

Caveat: I am NOT a lawyer. But I've cited my sources, look them up if you have questions.