If I understand correctly, if I am paying a floating rate like the 10y UST rate
When the market for interest rate swaps arose and developed in the 1970s, it developed some "market conventions". People follow them even through it's easy to think of other conventions that might be better in some way. (Think of it as th question, why do we drive cars on the right in the U.S., and on the left in the U.K.? neither way is better; it's just the way it happened historically.)
For interest rate swaps denominated in U.S. dollars, the market convention is that the floating leg pays 4 times a year, using "Actual days / 360" daycount convention, the rate is determined at the beginning of the period from a rate called 3 Months USD LIBOR. The fixed leg pays twice a year using a "30 / 360" daycount. I just googled and found some page that explains it nicely. For other currencies (e.g. EUR), the market conventions are not necssarily "better", but are "different".
We are, actualy,in the middle of switching from LIBOR (see 'LIBOR cessation') to a new index called SOFR. In the future the most standard USD IR swap is likely to have both legs paying once a year; and the floating leg will use SOFR reset in arrears (at the end of the period).
You can trade interest rate swaps with other terms and conditions (more on this below), but you will pay more bid-ask spread for the privilege. If you want a really unusual swap, then you will pay a ot of bid-ask.
I would pay the swap rate associated to the 10y UST according to the chart in the provided link
Not quite. The Chatham Financial page that you quoted lists a lot of rates.
U.S. Treasuries these are not swaps. Rather these are the yields on U.S. treasury debt. It's bad style not to provide a footnote explaining what these numbers are. Are they trying to confuse readers on purpose?
Swaps – Semi-bond I really don't like salespeople who mis-use what they think is "Wall Street jargon" in communications with clients, hoping to confuse and/or impress the latter. Semi-bond is shorthand for what I wrote above - 3 month libor versus semi-annual fixed. This is by far the most common kind of USD IR swap.
Swaps – Monthly Money this is jargon for 1-month LIBOR - see the page I cited above (again, going away to be replaced by SOFR within a few years).
Swaps are traded "over the counter" (OTC). This implies that there is no list of permitted swaps and "thou shalt trade no other". If someone wants to trade a swap that would, for example, pay 6M (rather than 3M) USD LIBOR and receive fixed, or for example, pay 3M USD IBOR and receive 6M USD LIBOR, etc, etc - it's easy to find someone to trade with and to see quotes for such swaps. It's just that the most commonly quoted USD swap rate is the fixed rate versus 3MO LIBOR.
If someone with tens of millions of dollars to spend, and the appropriate agreements in place, really wanted to trade a bespoke swap that you described - paying US treasury yield and receiving fixed - they'd probably be able to find some bank wiling to do it. But the bank would charge a lot for this work (by paying less fixed or receiving more fixed). I can't think offhand why someone would want to trade that.