I will answer the question about the "interest rate" meaning.
When people say "interest rates rise"; the root cause is usually federal reserve lending rate to banks as you said. The direct impact of other cause could be the inter banks lending rate. LIBOR is the most used indicator in the U.S. It is reported manually by every major banks once every day on the evening. It is reported in the 5 major currencies for seven keys maturities overnight/spot next, one week, and one, two, three, six, and 12 months. It is a kind of a volume-weighted average of the rates at which they borrowed and lent money to other major banks throughout the day. Hundreds of thousands of these transactions happens every days involving billions of dollar in cash from an account to another in both directions.
When these rates rise all other traded rates will ultimately rises.
Here is why:
Assume that banks can borrow from the U.S government from 1% for 1 year. And the next day the federal reserve (US gov) can only lend at 10% because they decided they want Americans to stop consuming so much and put some money in their saving account. I will stop the joke here.
In this scenario, the rates at which a company borrow money from a bank will also rise from 2% to 11%. If not let's borrow money at 5% from JP Morgan and lend in to Goldman Sachs at 8% they would be very happy it is way better than the 10% of the US gov. Pretty much anything that has time value in it will logically rise.
In this scenario your loans rates to buy a house should rise just as much from 4% to 13% a year.
You can usually decompose any interest rate as base rate + margin. Where base interest rate in the interest rate from someone less likely to pay you back margin > 0. If not an arbitrage is possible. Banks and federal reserve are (still) considered amongst the most reliable borrowers therefore they are used as a minimum base rate by anyone who is willing to lend money and all the other rates shift with them.
FYI: LIBOR (the inter bank reference rate I described on my first paragraph) has been subject to many scandals (banks deciding which number profits them the best based on their positions of the day and provide this rate as a base for everyone else). It will be deleted. People will start using different basis like SOFR.