In this video for smart cropping, the tool creates several new renditions that are crops of the original image.
At the moment, if I edit an asset and create a crop, it overwrites the original image. Is it possibly to create multiple crop renditions of the same image manually?
The non-destructive image renditions (through image modifiers) is a key feature of Image Server (part of Dynamic Media SKU). DM license is needed for the extensive support around this.
Just in case if you need to generate image renditions of different dimensions on the fly, there is a OOTB servlet "com.day.cq.wcm.core.impl.servlets.ThumbnailServlet" which use Layer API and generate the expected thumbnail on the fly.
For e.g. if user navigates to
then given servlet gets called and generate the thumbnail on the fly but, beyond this, DM licence needed for all advanced use cases.
Can you confirm if the Dynamic Media is enabled? if not use  to enable it.
In Assets (when Dynamic Media is disabled), the image edit operations (cop, rotation etc.) results into destructive binary edit. Where as in case of Dynamic Media enabled, the image edit operations are performed using modifiers and doesn't modify the binary. Refer for more details on DM.