You're definitions are very good, except a system is considered to have memory when its current output is dependent on future values of the input and output also. - Christen Juzeszyn
Correct, clear to me. - Ronny Wijaya
These definitions are both correct and simple to understand. I might change the wording "It doesn't take in..." to say, "It doesn't depend on...", but this is simply a matter of personal preference to wording then anything that is really incorrect. As mentioned above, you did forget to mention that a system with memory depends on the future as well as the past.
-Zachary Curosh
Like Christen I also noticed that you did not mention the fact that, for a system with memory the ouput y(t) depends on both past and FUTURE values of inputs. Also i just wanted to point out the fact that the term (t-1) represents a time "t" in the future not the past. :)
Vivek Ravi
