On Garrys mod i lag out at least every couple rounds. And some of those rounds im being slayed .... So my suggestion is to not ban right as they leave but give them a 15 minute return time. This has happened 3 times. I am banned for 60 days for this... I lag out try to rejoin and i dont have "evidence" of it. Please do this.
You haven't provided any evidence that your game crashed. Please provide the evidence, as stated in your ban appeal and then we could talk.
Go to: C:\Program Files (x86)\Steam\steamapps\YourSteamUsername\garrysmod (Just normal Program Files if you do not have a 64bit OS) and you will find a file with a name like: hl2_140108_crash_2014_1_11T21_51_53C0.mdmp
It'll be the one you sign up with. And usually I give multiple rounds for the person who leaves to come back, unless they are clearly intentionally leaving. Most other staff do the same thing.
The easiest way to solve this problem is to add the staff member that banned you to your friends on steam. It would look more probable to us to receive an IM shortly after we ban you than for you to wait a day or two and post that your parents told you it was bed time or that your GMOD crashed.