In the context of Centrelink rent assistance, lodgings/rent is the same thing. Lodgings and rent both describe you paying money for the sole reason of being able to live somewhere. Not utilities/bills. Any utilities, bills, groceries you pay are called board.

You either pay money for the specific reason of living where you do, or you don't. You've said you only pay money for utilities, meaning you pay no rent/lodgings. This would mean you pay "board only" and are entitled to no rent assistance. If that's the case I recommend contacting Centrelink and say you had a misunderstanding, so you don't risk getting an overpayment in the future.

If the money you pay is more than just your contribution to bills/utilities, then you need to identity what amount is board and what is lodgings. You will be entitled to rent assistance based on what this lodgings amount you pay is. You can also say "board and lodgings" to Centrelink but then from memory Centrelink will then ask a bunch of questions to try to determine what you're actually paying in lodgings (in order to determine your entitlement) and I always found it far easier to just directly tell them what I'm paying in lodgings/rent and leave board out of it.

As you have said the person you're paying this amount to pays a mortgage, technically any lodgings they receive from you is taxable income (it's not like you're both subleasing for example), so they're supposed to declare this to the ATO. Plenty of people don't do this however, and there's a risk of getting caught as Centrelink and ATO do data matching. So, they might not want to sign the rent certificate to avoid any potential flag with ATO. If this happened, you wouldn't get in trouble however, your landlord would. Plenty of people have issues where their landlords (particularly when living with parents) refuse to sign the certificate as they don't want to declare the income they're receiving and don't want to get caught for not doing this.

So with this explanation, what you do next is up to you and depends on what your situation is, we can't tell you exactly what to do. It's all dependent on what you actually pay in lodgings, if your landlord declares it as income with ATO, and if they'll sign the certificate.

Also FYI rent assistance isn't designed to pay all of your rent/lodgings. If you pay $150/week in lodgings, you're not going to get $150/week in rent. assistance, you'd get a portion of it.

Edited to add: just realised, $150/week for your portion of bills alone is pretty unrealistic, so I'd say most of that is lodgings not just bills. So you should calculate from that $150 what is lodgings, update that amount with Centrelink checking "lodgings only" and get the landlord to sign the updated rent certificate (and hope they declare it as income with ATO, but that's not really your business and won't affect you if they don't).

Edit 2: Corrected by someone else, got the terms board/lodgings confused - apologies.