I read online that the Arduino wall adapter has a 9 V, 650 milli amps output.
There is no such thing as an "Arduino wall adapter". However, there are wall adapters that work with Arduino.
Current ratings are a matter of "source" and "drain". The power supply current rating is the amount it can "source." The load (whatever you attach to the supply) will draw from that source. For example, the Arduino will draw around 40 or 50mA when powered through the barrel jack. This means it will only draw up to 50mA from the source.
In theory, It doesn't matter what that 9V source is rated at, as long as it is more than 50mA. In practice, you don't want to go way the amount of current you need, because regulators generally have a minimum load requirement, blah blah blah. Don't go over 2A unless your project has a specific need.
I have both an 8v, 650 mA, and a 9v, 400 mA wall adapter from other electronics. Would either or both of them be sufficent to at least run the Arduino?
You should be able to answer your own question now...