The record seems to be pretty solid that Col. Joe Rickey really did invent this drink in Washington DC. It was probably during one of the scorching summers as this drink has a cooling effect like few others (Col. Rickey claimed that the absence of sugar was a big reason for this). This is a go-to hot weather drink. Use large ice cubes or smaller ones if the heat is not oppressive.
The original Rickey (also called a Joe Rickey at the time) was made with whiskey (probably bourbon). Afterwards a number of variations popped up with the base spirit used as a part of the name. One variation, the Gin Rickey became hugely popular, totally overshadowing the original whiskey version that Col. Rickey had invented a decade before.
If you want a cleaner presentation, strain the lime juice to remove the pulp. With so much carbonated water the pulp will turn into "floaties" in a Rickey. I actually prefer this so I don't strain the lime juice.
Usually I advocate club soda for drinks like this because I have found some mineral waters lacking in carbonation. But the mineral flavors really do go well with the citrus and whiskey in this drink so I have specified it here. Also, you are using quite a bit of mineral water so even with so-so carbonation it should fizz up nicely.
Your lime juice has to be fresh from a fresh non-funky lime. There is no sugar to cover up off flavors.
And please do try a Rickey with rye whiskey. I find these especially refreshing with the rye's depth and spicy notes.