Thanks for all of this information, very helpful.
After a little research my understanding of the gift tax is that it must be reported if it's over $17k/$36k but no taxes need to be paid unless somebody has gifted over $12.9 million in their lifetime? So not something to worry about for most people.
It's $18K per person per year this year; the number goes up with inflation.
And it's $13.61M lifetime for people dying this year. If the law doesn't change, that will drop to about $7M in two years. If she passes away in 2026 with an $8M estate, her estate would owe federal estate tax (because $8M > $7M). And there could be state estate or inheritance taxes at lower numbers. But again, this applies to very few people.
Sorry if I'm being naive here, but isn't it a somewhat regular thing to negotiate on the price of a house and try to find a sweet deal? And if you make/find an undermarket deal Uncle Sam considers it a gift? Like what if I find a run down property that's not on the market and I find the owner and they agree to sell it to me for cash for cheap because they're in another state and don't care about it or something... would they have to report this as a "gift of equity"? Doesn't seem.. right. Also, grandma (or her estate) in this situation will wind up with more money in the end than a traditional sale at the market price, even if the "sale price" is $100k less.
It is regular to negotiate on the price, but most transactions between strangers are not at below market values because most sellers will market the property and take the best deal they can get, not the first deal that comes along. Also, most sellers, even if they like their buyers, won't give their buyers a deal just because they like them. Neither of those apply to your grandma situation AFAICT.
As mentioned, it would be reasonable for your grandma to discount the sales price from market by the selling costs she would avoid...but if you discounted by much more than that she and you may run afoul of the IRS. I really doubt you can come up with $100K of selling costs she's avoiding. Will you be caught? Probably not, but it's up to you if you can sleep at night in that scenario. It's also not that hard to do it by the rules - see next paragraph.
As alluded earlier, you can probably structure an installment deal where your grandma sells it to you for $472K, $400K this year and $36K a year for the next two years, and then she chooses to forgive whatever amount she feels like up to $72K. Probably no capital gains taxes, no gift tax return, and you effectively get the house for $400K. There's the niggling detail of interest at the AFR for those two years, but that shouldn't be much and you could probably massage the deal to handle that aspect also.
Given the unusual situation and your being new to the tax rules, I'd strongly suggest getting in touch with a local tax preparer who can guide you and your grandma on the numbers and the rules.