Money invested vs increased valuation
I don't know the name of the term you are looking for but the ratio you are looking for can be calculated as the volume of trade data is usually available for each stock. A similar metric that you may like is Volume-Weighted Average Price which takes into account the volume of trade.

Is this suspicion accurate?

It's very subjective but your suspicion seems to be based on your hypothetical construct which needs further discussion. You say 'I really want to buy a share..'. Now why would that be? For a rational investor, it will be when she sees it's value to be \$11 rather than market value of \$10.
When enough investors are there in the market that think that the value of the share should be high then the price of the stock goes up (and there's nothing wrong with it), which reflects, in a way, the collective sentiment about the stock.
In the above reasoning, we consider the market as a non-cooperative non-atomic game, in which single player's decision has no influence on market outcome. Therefore, an individual is not pushing the price of the stock but rather the general consensus that the stock is going to giver higher returns.
So your suspicion is correct when these at least one of the following assumptions are not be valid: (a) investors are not rational (domain of behavioral finance); (b) game cannot be considered non-atomic (there are big players that can alone move the market - almost always true); and (c) there is collusion among investors to bid the price of a stock up.
So yes, for some stocks, to some extent it is perfectly possible that some investors bid the price up to the level that it doesn't reflect the real valuation. This is more common with penny stocks which is why there are more restrictions/regulations around them.