How can I find out if an Item was listed / Relisted via the website or my application via API calls?
You can use the either of the tags UUID, SKU or ApplicationData to identity if an Item was listed using the API as these are not available via the website.
You can pass in anyone of these tags in the AddItem / RelistItem API call request to establish that the item was listed via the API, since the SYI flow does not offer these tags:
UUID SKU ApplicationData
When you make subsequent GetItem calls, you will be able to get these fields back in the response.
For Relisting the Item, if you do not make any changes to the item, the ApplicationData and SKU will be carried over from the original item, however UUID will not be carried over unless you specify a new UUID. Hence, of the three, UUID is the best alternative, because it prevents duplicate listings also, besides helping you identify that it was listed using the API.
As for MyeBay, there is no way to find out how the item was listed / relisted.