You're overthinking. First middle last is fine.
The W-9 will be used by your employer to collect your SSN, which they will use on tax reporting documents. IT's more important that the employer get your name right when they submit their tax documents to you at the end of the year (with a copy to the IRS). The W-9 doesn't actually go to the IRS.
What is more important is, what kind of job is asking for a W-9? If you are going to be a regular employee with a paycheck and a W-2, the employer should be using an I-9, not a W-9. W-9 is more commonly used if you are going to be treated as an independent contractor or self-employed. This may be appropriate, depending on your duties and your relationship to the payer, but being considered self-employed has important tax consequences. You must pay your own estimated taxes 4 times a year since you don't have withholding, and you will likely owe more taxes than if you were an employee doing the same work (often, contractors ask for a wage bump to account for this).
You may want to look at this to start,
https://www.irs.gov/newsroom/understanding-employee-vs-contractor-designation
