It is all games. Do those that count their interest and dividends also count appreciation in equities as additional savings? If not, why not?
No. Appreciation of equities is a reflection of the increase in current value of an asset, not a receipt of new capital.
Dividends are not new capital. They are a distribution of the capital that you own. If you reinvest the dividends then you are exactly in the same spot that you started with. You are taking dollars out of one pocket and putting it in another pocket. That is why stocks drop when they have a large dividend payment. The company's value goes down by the dividend that they paid out. The company didn't magically create more value by doing a dividend.
That is not the same as receiving cash in the form of dividends/interest, which is income. I dont see why you wouldn't be able to count those as savings. I agree in reality for most people it's not a large enough number to move the 'savings' needle. But some people absolutely get large ammts of dividends/interest...some people even live on it as their income. For those that don't, I don't see why if you get $1000/month in interest and don't spend it it's any different than saving $1000 out of a paycheck.
An easy way to look at this is rental property. The rent i get is income. An increase in the value of the house is not, it's a capital gain. I think you'd be hard pressed to find many people who think the rent received on a house isn't income, or that if they net $1000 per month from a rental and don't spend it, that it's not savings. Interest or dividends are no different, they are cash received as a product of an asset you own.
Again, it does not matter what you count as long as you are comfortable with the calculation. You should not use the Shockingly Simple Math to calculate your FI date as that calculation already has a 7% return built into the calculation. The calc assumes that your investments will grow with value. So if you strip off the interest that is suppose to be reinvested, and you count it as savings then you are double counting the dividend or interest. The calc will be off. No big deal, just understand that is how it works.
It also seems strange that you would count dividends and interest, but not count asset appreciation. Berkshire Hathaway has never done a dividend. To say that an interest or dividend paying investment is better or increases your savings seems a bit confusing. Your desire for dividends or interest may skew you away from investing in appreciating assets that don't have a dividend yield.
Rent on a house is the return on investment. The Calcs have a return already built in. If you turn around and spend your returns then you are treading water. The goal is for the returns to compound. I am not saying that you are wrong, just that when we talk about savings rates, we are talking about putting dollars in a pool to compound. If you are taking the dollars out by counting dividends or interest, then you are not allowing them to compound. If you reinvest you are getting a double dip.
Do you go to you 401k, IRA, mutual funds, and other funds and figure out how much is dividends/interest and add that back to your income? Curious.