"Inflation is not about prices" is the silliest thing I've ever heard.
You might as well say that gravity has nothing to do with up and down.
The only thing anyone MEANS when they say "inflation" is that it's short for "price inflation".
But to deal with your point directly: if the U.S. Treasury literally printed a quadrillion dollar bill but then gave it to someone with a contract saying that they agree never to spend it, it will have zero effect on anything. And if people get dollars and never spend them, it's the same effect in practice. Or if someone with a quadrillion-dollar bill gives it as a gift to someone else who then later gifts it, and that's all that ever happens… again, no effect on inflation.
Inflation is one thing and one thing only: it's when people who make the decisions about setting prices for things choose to inflate (increase) the prices and that this happens on a noticeable system-wide scale. If the people who set prices chose not to change them, there would be no inflation, period. The interesting thing in studying inflation amounts to studying what patterns correlate with people making the decision to increase prices. And yes, knowledge that buyers have more dollars to spend is a factor that could (and does) influence those decisions on setting prices.
The same thing happens with "demand" (so, that professor I summarized above is still not quite right when he asserts that demand and spending is what causes inflation). Sellers can very well (and do often enough) keep prices unchanged even when demand is high and everything just sells out quickly. The result of that isn't inflation, it's shortages — unless the shortages somehow motivate people to just produce more — in which case increased spending just leads to increased production and consumption without inflation or shortages.
Shortages don't mathematically cause inflation. Inflation ONLY EVER happens if price-setters choose to increase prices. Nobody is EVER forced to increase prices. You just have the ramifications of doing so or not. Maybe keeping prices the same means less profit. Or maybe it means going bankrupt.
And yes, any one decision to change a price has an impact on other people who may choose other prices. That's why all the interacting decisions from all the actors adds up to patterns we can potentially (but always imperfectly) predict.
This is human beings making decisions and taking action in a complex game we play around money. Yes, there are mathematical aspects to it. But it's not some pure math abstraction. And relying too heavily on math abstractions is one of the deepest flaws in the whole field of economics.