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The terms "partition" and "volume" can usually be used interchangeably. Technically there's a partition type that can contain multiple volumes, but primary partitions, which are used most commonly (and exclusively on GPT disks) can only contain a single volume.
In terms of your error, that's usually caused by running the "assign" command before you've run both "select disk x" and then "select partition x", both times substituting a number for X. Did you perhaps only enter "list partition" beforehand to determine the partition number of interest and forget to enter the command to actually select it? Once you get your letter assigned, you would exit diskpart and then enter "chkdsk /r x:", again substituting the assigned drive letter for X. You may also find it easier to do all of this from within the Rescue Media environment, since not all partitions can be scanned while Windows is booted.
If you're still having trouble, can you post either a screenshot or the text output of your entire diskpart session, including the output from the "list partition" command after selecting the correct disk?