Disk space full during insert, what happens?
The actual query wasn't executed in an explicit transaction though,
  can that explain the existence of the target table?


Yes, exactly so.

If you do a simple select into outside of an explicit transaction, there are two transactions in autocommit mode: the first creates the table and the second fills it up.

You can prove it to yourself this way:

In a dedicated database on a test server in simple recovery model, first make a checkpoint and ensure that the log contains only a few rows (3 in case of 2016) related to checkpoint. Then run a select into of one row and check the log again, looking for a begin tran associated with select into:

checkpoint;

select *
from sys.fn_dblog(null, null);

select 'a' as col
into dbo.t3;  

select *
from sys.fn_dblog(null, null)
where Operation = 'LOP_BEGIN_XACT'
      and [Transaction Name] = 'SELECT INTO';


You'll get 2 rows, showing you had 2 transactions.


  Are the assumptions I sketched here correct? Is this a likely scenario
  to have happened?


Yes, they are correct.

The insert part of select into was rolled back, but it does not release any data space. You can verify this by executing sp_spaceused; you'll see plenty of unallocated space.

If you want the database to release this unallocated space you should shrink your data file(s).