You are reading the IRS instructions correctly. Your employer should put on your W-2 in box 12 with a code of W only the amounts that the employer actually contributed to your HSA plus whatever amounts you contributed through your employer by means of payroll deduction.
Your employer should not add anything else to the code W amount. Insurance premiums paid by the employer are normally listed with a code of DD in the same place.
Your W-2 is incorrect and you need to ask for a corrected one. This error on the W-2 will possibly cause you to appear to have contributed an excess of contributions to the HSA as well as cause an incorrect form 8889.
Do not file until you get a corrected W-2. It is important to get the corrected W-2, because this will force your employer to also send the corrected W-2 to the IRS.
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