Can I use Unity (free edition) to make a non-commercial software?

You would need a plus / pro license. The Eula is actually pretty clear. Read about how you finances are measured.

If you are part of the company you do the work for the first point is relevant to you:

if you are a Legal Entity using the Unity Software (other than to provide services to someone else), your Total Finances are your gross revenues and/or funding (no matter what the source), or, if you are a Not-for-Profit Entity, your budget for the entire entity (not just the department using the Software),

If you are not part of that company but you do the work for them, the second point is relevant:

If you are an individual or a Legal Entity providing services to a third party, your Total Finances is deemed to be your customer or client’s Total Finances.

Just in case you are an individual and not providing your work for someone else (which clearly isn't the case here) the thrid point would apply:

if you are an individual using the Unity Software, but not providing services to a third party, your Total Finances are the amount generated in connection with your use of the Unity Software. In this case, your Total Finances would not include amounts you generate from other work (for example, if your day job is as a zookeeper).

However since you are not an individual and you actually do provide services to someone else your finances are your customer / clients finances. Since he's over 100K you need at least a plus license. If he is over 200k you need a pro license.

Keep in mind that the minimum committment period is 24 months. After that period you could cancel anytime. Note that builds created with Unity can be sold and used as you like even after you cancelled your Unity license. However you are no longer allowed to use the Unity editor in this case. So if you just create a product which you do not need to modify, there's technically no need for a license after that. However in many cases you need to do some minor changes / bug fixes in the future in which case you would need a license.

Final note: This is not a legal advice. This is my interpretation of the Eula. I'm not a lawyer so don't take my word on that matter. If in doubt contact the Unity sales team or consult a lawyer. Also for future readers, keep in mind that the eula and terms of services can change in the future and the information provided here might have become obsolete.

Thanks, I think your response makes perfect sense.

I still not sure if I can prepare a Proof-of-concept before actually being able to use it or I need only when deploying it (probably the former as it the EULA refers to the "USE" of the editor software). I'm seeking legal advice withing the company and submitting my questions to Unity support, but your explanation made it a lot clearer.