Your understanding is correct. About your question regarding the effects of contributions being made via payroll deduction, it doesn't matter how you make the contribution. Just when, and how much you are allowed to contribute. What makes a difference in your taxable income is how much you contribute to the HSA, with in what is allowed per person, per age and income. For example, you can't contribute more than what you earn. You can contribute an extra $1000 when you are 55 and over, etc.
If your husband is only contributing a fraction of what is allowed, and you would like to contribute more, you can make the contributions manually. You can even wait until you know what your tax liability will be and make the additional contribution by April 15, Tax Day, same as with an IRA contribution. https://www.irs.gov/pub/irs-pdf/p969.pdf
