Halifax mortgage overpayments - Reducing term, monthly payments - Explanation needed

I've been looking to understand how mortgage overpayments effects the mortgage term or the monthly payments. Not sure if this is specific to Halifax, but they have this policy that confuses me a lot. They won't reduce the term automatically after an overpayment, instead, they say that they will adjust your monthly payment when the monthly payment recalculation happens.

Here is a quote from their overpayments calculation site.

If you’ve got a Halifax mortgage, overpayments won’t automatically reduce your mortgage term or your monthly mortgage payment, but could save you money by reducing the amount of interest charged. When the next monthly payment recalculation happens, for example at an interest rate change, the monthly payment will be calculated using the existing remaining term of the mortgage. If you want to reduce your term you will need to speak to one of our mortgage advisers to discuss your options.

The situation we are in currently is the following:
Fixed rate of 1.9% for 5 years (around 4 and half years left). The overall term left is 19 years and 6 months.

By reading the other threads on similar issues, what I understand is that to reduce the term of the mortgage, they will need to run affordability checks and so on, similar to applying for a new mortgage. That bit I understand, because it is essentially changed terms of the contract.

But what I don't understand is the two points.
1. What other events causes the monthly repayment to be recalculated? They have provided the interest rate change as an example in their site. For example, does a lump sum payment cause a recalculation? Or does the monthly payment remain the same?
2. How this effects the overall amount of interest I will have to pay even if I use overpayments.

For example, let's say our principal amount is 100,000£ for now and we have 10% overpayments per year. The current monthly payment will be 511.73£ per month of which around 159£ are interest payments. If I now pay 10,000£ as an overpayment, will they reduce my monthly payment to make sure we will be owing money until the end of the 20 year term? Or will they keep the monthly payment as 511.73£ where a smaller portion of it will be interest payments, in this case, I expect it to be around 149£?

If they do the latter (keeping the monthly payment the same), that means that by end of year 16, we will have payed the mortgage completely assuming that the interest rate is the same (which is a big assumption, but for the sake of this argument, lets assume that).

In other threads, people also mentioned things such as re-adjusting the monthly payments to make sure we pay the same amount after they change the monthly payment. Which confused me even more because if an overpayment causes the monthly payment to be adjusted, they more overpayments will cause the same. and so on.

Basically, if someone could answer my questions and clarify this whole mess a little bit for me, I will highly appreciate that. As last time I tried to get the answer to these questions from Halifax, they were not helpful at all.