In the US we usually don't use "till" in this sense. We call it a "cash register." The "cashier" is a person who takes your payment in a store, whether by cash or credit or debit card, especially in grocery, drug, and other similar stores where you pay for your goods all in one place, rather than buying them in a the department you chose them from. A"sales clerk" would be the person who handles the transaction in these cases. The location for payment to a "cashier" is typically called the "check out" or "check out counter" or the "register." If there is a sales clerk,,we'd usually use these same words with the "counter" probably being the mist common.

The person who takes your money is the 'cashier', and the 'cashier' then takes the money and puts it in the 'cash register' or 'till'.

The descriptions of these occupations may differ slightly in the English speaking world. For example, in Australia the person who takes your money at the supermarket is the "checkout assistant" who puts the money in the till or cash register. In a department store the sales assistant performs this task. Of course, these people are a dying breed as we move to cashless and automated checkouts where the customer does all the work and a member of staff hovers about to reset what are called in Australia at least, the Point of Sale Terminals or just "terminals", and to make sure no-one sneaks out without scanning their purchases.