The Toddy is presumably an American drink that predates the cocktail by several decades, with the earliest known references being in 1750 (July 1750 edition of The Boston post-boy - need to find reference). A Toddy is almost always a hot drink, if made cold, it would be hard to distinguish it from a Sling (English), and if it is made with a lemon twist, it would be difficult to differentiate it from a Skin (Irish).
Traditional recipes consist of a spirit, mixed with hot water and a sweetener of some sort. Sometimes it is garnished with a lemon twist or even a lemon wedge, and a sprinkling of nutmeg or cinnamon. The fact that it is often referred to as a "Hot Toddy" could be an indication which supports the notion that it could be served either hot or cold.
You could consider a toddy as a hot variation of punch, and like punch, toddies could be commonly served to a group instead of individuals. While it could be brought to the table in a bowl similar to how punch would be served, it would cool quickly, which made serving it in an enclosed porcelain pitcher or "Toddy Jug" much more appropriate. This would be much the same as a carafe of coffee might be served to a group today.
- 2 ounces Spirit (although over time Whiskey has become standard)
- 3/4 ounce Sweetener (Honey, Simple Syrup, or sugar)
- 2 ounces water (normally boiling hot)
Optional: garnish with a lemon twist, or thin wheel of lemon.
If you try to get a clear definition of the Toddy by using the Jerry Thomas 1862 Bartenders Guide, you will find it more than a little confusing. He lumps the Toddy and Sling together, appearing to indicate that both can be hot or cold, with the only difference being a Sling is server with a grating of nutmeg.
In this section he lists recipes for:
- Apple Toddy: Sugar, cider brandy, 1/2 baked apple, boiling water, garnished with grated nutmeg.
- Brandy Toddy: Sugar, brandy, water, ice. And indicates that a "Hot Brandy Toddy" would omit the ice and use boiling water.
- Whiskey Toddy: Sugar, whiskey, water, ice.
- Gin Toddy: Sugar, gin, water, ice.
- Brandy Sling: "...same ingredients as brandy toddy, except you grate a little nutmeg on top."
- Hot Whiskey Sling: Whiskey, boiling water, garnished with grated nutmeg.
- Gin Sling: "...same ingredients as gin toddy, except you grate a little nutmeg on top."
From this it sounds like all that differentiates a Toddy from a Sling is the addition of grated nutmeg, except that the Apple Toddy includes nutmeg. It is also strange that the Hot Whiskey Sling doesn't include sugar.
Fermented Coconut Flower Sap in Guam
An extremely early use of the term "Toddy" when referencing a beverage, comes from “A New Voyage Round The World.” (Volume 1), by Captain William Dampier. It was published in 1703, and is an accounting by William Dampier regarding his ocean voyage to Guam in 1686. In this he provides details of the Coconut Tree's found there, and how the locals obtain a liquid from the tree referred to as "Toddy" (also referenced as "Toddi"). 
Here is the full text of that accounting, presented in the form it was published. You will notice the use of "ſ" scattered throughout it. This is known as a "long-s", and should not be confused with an "f".
Toddy and Arack Liquors made of the Coco-Tree.
Beſide the Liquor or Water in the Fruit, there is alſo a ſort of wine drawn from the Tree called Toddy, which looks like Whey. It is ſweet and very pleaſant, but it is to be drunk within 24 hours after it is drawn, for afterwards it grows ſowre. Thoſe that have a great many Trees, draw a Spirit from the ſowre Wine, called Arack. Arack is diſtill’d alſo from Rice, and other things in the Eaſt Indies; but none is ſo much eſteemed for making Punch as this ſort, made of Toddy, or the ſap of the Coco-nut Tree, for it makes moſt delicate Punch; but it muſt have a daſh of Brandy to hearten it, becauſe this Arack is not ſtrong enough to make good Punch of it ſelf. This ſort of Liquor is chiefly uſed about Goa; and therefore it has the Name of Goa Arack. The way of drawing the Toddy from the Tree, is by cutting the top of a Branch that would bear Nuts; but before it has any Fruit; and from thence the Liquor which was to feed its Fruit, diſtils into the hole of a Callabaſh that is hung upon it.
There apparently is a similar accounting recorded as early as 1655 in "Voyage To East India." by Reverend Edward Terry, but we have been unable to locate this as a manuscript to use for reference. 
These descriptions of gathering the liquid from the coconut palm use the terms "wine" and "distilled" in their descriptions, but without an actual indication that it is either aged for fermentation, or even mechanically distilled into a true spirit. Apparently "Palm Wine" as gathered from the coconut palm, is such that it begins noticeable fermentation almost immediately, and while not very strong, will provide a certain level of alcoholic content. It can fairly quickly yields an aromatic wine of up to 4% alcohol content. In reference to the liquid gathered from the coconut palm, "Toddy" refers to the flower sap gathered from the tree. As indicated, it ferments quickly, but also has a short shelf-life of about 24 hours.
In this case, the word "Toddy" would be probably be derived from the Hindi word "tari" (palm sap), with the Hindi "r" sounding similar to the English "d".
In 1871, the New York Times includes a short recounting of how "Toddy" might refer to a Scottish incident where "Tod's Well" was providing the citizens of Edinburgh with "aqua" (water)... but with the wink-wink, nudge-nudge, that the Scotish whisky trade often invoiced whisky as aqua. 
The New York Times - Sunday, January 1st, 1871
The Origin of the Word “Toddy.”—Here was, and is, a well called “Tod’s Well,” whence Edinburgh in the bygone days, when water was a scarcer commodity than it out to be in any well-regulated municipality, supplied the city with as much of the pure element as sufficed for that primitive and unsanitary time. It may be mentioned that, as aqua vitae in Latin, eau de vie in French, and usquebae in Highland Gaelic, severally mean the “water of life,” so “toddy,” of which the Scotch at home and abroad seldom lose the love or the flavor, seems, if we may believe an allusion in Allan Ramsay’s poem, “The Morning Interview,” to have originally meant water without any whisky in it. Speaking of the adjuncts to the breakfast-table, the tea brought from the Eastern, and the sugar from the Western hemisphere. Ramsay says that Scotland brings to the feast “no costly tribute,” but
And explains the passage by the statement in a foot-note, that “Tod’s Well supplies the city with water.” The custom in Scotland, in the whisky trade, to invoice whisky as aqua, lends strength to the supposition, and tends to disprove the allegation of the dictionaries that the word “toddy.” Is derived from India, where it signifies a kind of arrack. – All the Year Round.
In 1801, we find in "The American Herbal or Materia Medica" by Samuel Sterns , a recipe listed for the Toddy:
This liquor is prepared by adding to three half pints of water, one of rum or brandy, a little ſugar, and after ſtirring, a little nutmeg.
This appears to be the oldest recipe for a constructed beverage going by the name of Toddy. Note that this recipe is not for a "hot toddy" but presumably a cold (or at least room-temp) one instead.
A key ingredient in a toddy, is usually some sugar as a sweetener. If you were making a hot toddy, this sugar would readily dissolve in the hot water, but if you were making a cold (or room-temp) toddy, the sugar would need a little more encouragement to dissolve, especially when using chunks of loaf sugar. In the early 1800's, a common tool for this was a small wooden mortar, typically with a flattened end, which would be used to crush and blend the sugar with the water and spirit. The "Toddy Stick" would be handy to use in this fashion with any cold beverage which relied on dissolved sugar.
Over time, sugar syrups, and flavored syrups would replace the use of loaf sugar for sweetening mixed drinks, especially when "ice" started to become popular around 1830, which made sugar even more difficult to dissolve. The Toddy Stick would be repurposed for instead muddling fruits, mint, or other ingredients when used to make drinks. This would eventually take on the name of "Muddler" instead of "Toddy Stick".
- Hot Toddy (Wikipedia)
- The Official Cocktail of Winter: The Hot Toddy (TheDailyBeast.com)
- How Well Do You Actually Know the Toddy? (Punch.com)
- The Real Hot Toddy (Liquor.com)
- Toddy and Arack Liquors made of the Coco-Tree "A New Voyage Round The World.", Captain William Damiper (1703).
- Hot Toddy Drink - Far East Of Scotland Long Before The Cocktail via BarinaCraft.com, Jan 10, 2022
- Toddy and Palm Wine (appropedia.org)
- The Origin of the Word “Toddy.” The New York Times - Sunday, January 1st, 1871
- Toddy The American Herbal or Materia Medica, Samuel Sterns (1801)