• Looks like you have a standard "adjustable" bottom bracket.

    I don't know what you mean by "lock nut".

    The non-drive side cup is the adjustable one and you will need to "fanny around" with it before tightening the lock ring.

    The tool you linked to does have a "pin spanner" on one end, but it isn't adjustable so you need to check that it will fit your cup. This is an example of the adjustable type.

    This page explains things in more detail.

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