Oh you really thought you had me with that one just yelling if this is true then the other thing must be true. Let me explain your mistake. You are treating all foreign support as identical. The books I cited do not do that and if you had actually read them, or had any clue regarding the subject you're attempting to debate, you would know it.

Tyrone Groh writes about the principal-agent relationship. The patron directs the use of force by the local actor. Not just sells them some hardware. The US supplies Ukraine and also sets the boundaries of what Ukraine can and cannot do. No ATACMS strikes inside Russia without permission. No F-16s until Washington says yes. Ukraine is operationally independent on the ground but strategically dependent on American will. That is what makes it a dynamic of a patron and a client.

On the other hand, Russia sells Iran drones and Iran pays for them, but Iran then uses those drones however it wants against whoever it wants. Russia does not direct Iranian foreign policy, nor does Russia tell Iran when to attack Israel or which militias to fund. Iran does that entirely on its own for its own reasons. That is not a relationship where one party directs the other. That is an arms deal between two sovereign states who happen to share some enemies.

Russia and Iran are business partners while the US and Ukraine are in a patron and client arrangement where the client cannot survive without the patron. Those are different points on Mumford's spectrum.

You also said books are just opinions and do not change definitions. That is the most anti-intellectual thing I have read all day. The dictionary is a starting point. The books are the analysis of how that definition actually applies to real conflicts. You cannot just scream the dictionary says support and ignore four hundreds of pages explaining what kind of support and direction actually constitute a proxy war. That is like saying a medical textbook is just an opinion on what a heart attack is.

You asked which is it are the books right or wrong. The books are right and you are dismissing them in favor of a superficial definition you googled. The US is fighting a proxy war through Ukraine because the US sets the strategic parameters. Russia is not fighting a proxy war through Iran because Iran acts independently. Both conclusions come from the exact same scholarship. You just cannot accept that because it complicates your little bumper sticker worldview.