A Bond's Asset Swap Spread is defined as the difference between Bond's Yield and the Risk free rate.

Then I was told that, the Present value of the Bond's Asset Swap Spread is the difference between this Bond's price and a similar risk free bond's price.

difference between Bond's Yield and the Risk free rate

Definitely not quite.

There are many different variations of comparing bond return to risk-free rate (swaps or treasuries) - e.g., option-adjusted spread (OAS), Z-spread (zero-volatility spread), coupon-adjusted spread... if you bring up YAS screen on Bloomberg terminal, you also see I-spread, G-spread, etc. To get a feel for any spread, you should calculate it manually (as opposed to looking at a Bloomberg screen) for many examples of bonds.

Each one has its own methodology. For most bonds, all these numbers are not going to be very different, but that difference is significant.

An asset swap spread is the spread over LIBOR (soon to be replaced by SOFR) if you swap your bond's fixed cash flows into floating cash flows; LIBOR + asset swap spread. To get a feel for it, you should calculate it manually for a few examples, including bonds with large coupon, in the middle of the coupon period, amortizing bonds, cross-currency asset swaps where the other currency pays fixed or floating, etc.

Most most bonds, the asset swap spread figure is going to be in the same ballpark as the bond's yield minus swap rate, or the Z-spread, but not exactly.

the Present value of the Bond's Asset Swap Spread is the difference between this Bond's price and a similar risk free bond's price.

Very approximately, this is true for any spread mentioned above. You have to assume too many things for this to be exactly true.

Your bond trades in the market at some observable price. Suppose that you discount the remaining risky cash flows of your bond with risk-free rate and added them up. The sum would be the fair value of a non-existent risk-free instrument having the same cash flows as your bond. Your risky bond trades at a discount to this value. But looking at how much greater this value is than the observable bond price isn't very insightful. Instead, you look at how much extra return you get every year to compensate you for holding this risky bond. That can be calculated in many different ways whose reuslts will be in the same ballpark.