[RPG] Am I correctly understanding how the Smite spells work

The reason things are worded so badly lies in the 3.0->3.5->PF transition. See below for that! I believe chill touch is still resolvable using the RAW, though.

Chill Touch

The answer to the chill touch question lies in the rules for holding the charge:

Touch Spells and Holding the Charge: In most cases, if you don't discharge a touch spell on the round you cast it, you can hold the charge (postpone the discharge of the spell) indefinitely. You can make touch attacks round after round until the spell is discharged. If you cast another spell, the touch spell dissipates.

Most touch spells discharge after a single successful attack, but chill touch makes it clear that this isn't the case. Instead, you get one touch per level. So the quoted rule allows chill touch to be used across multiple rounds; there's no need for the spell to explicitly state it. (Though it probably should, just for clarity.)

Willing targets

The rules do explicitly state that you can't hold the charge on multiple targets, so no help there! That seems to be the simplest solution, though.

As an aside, an effect that changes the range of touch spells would "unlock" the full number of targets, so there is a technical difference between the current state and your alternate wording. :P

A little bit of history

So why are the RAW in this state? Well, I did a bit of digging, and this thread was very helpful, especially this post by Hypersmurf. (Who I remember from when I posted in the enworld rules forum far too often -- thanks Hypersmurf!)

1. In 3.0, you could touch 6 willing targets as a full round action, or one as a standard. There was no restriction on touching all the targets the round the spell finished -- in this context the 6 target limit played well with multi-target spells.

2. In 3.5, the Magic Overview section was changed: the limit of 6 was removed, but the requirement of touching the targets in a single round was added. However, the rules for touch spells in combat left the old restriction of 6 targets in place. The text here directly contradicted that of the overview section. If you assume that the wording of the overview section was what the designers intended, this would still cause no problems with touch spells targeting multiple willing creatures.

3. In PF, both limitations are included in the overview. I would assume that someone was trying to reconcile the contradiction mentioned in 2, without realizing that it was a holdover from an intended 3.0->3.5 shift. The result? Accidental nerfing of multi-target touch range buffs.

Likewise, the wording of chill touch has been essentially unchanged since 3.0. It's been edited a little to make the wording less clumsy, but not updated to account for the changing touch attack rules. This explains why it doesn't explicitly call out that it carries across multiple rounds.

Edit: This answer predates several rulings and clarifications made by WotC and Crawford in particular. I'm leaving it in place for historic purposes, but it's no longer a particularly useful answer.

Strictly speaking, there is no clear interpretation. All three cases are justifiable. Also note that 5e discourages literal "rules as written" meanings. As the designers have repeatedly said: "rulings, not rules." The rules were explicitly not written to be scrutinized as a lawyer scrutinizes the law, so we should not be surprised when the end result of "it's ambiguous" is what we find.

Firstly, "natural" melee weapons are, as far as I'm aware, considered melee weapon in 5e. [ See also.] There is no distinction between a mace and a hoof as far as "counts as a weapon" is concerned in 5e. I don't know if this is explicitly stated anywhere (I thought it was) but Unarmed Strike is explicitly listed as a weapon on the weapons table, and it's strongly implied since all monster stat blocks say things like "Bite Melee weapon attack: [...]". As far as I can tell, if you make an attack with it, it's considered a "weapon" in 5e. Something is a weapon if it's used to make an attack, then, not because it's got a weapon tag on it.

You could argue a Case 1 by saying that find steed only modifies the target of the spell. The spell still refers to "you," so even though it effects your mount, that extension does nothing. In other words, you argue that for Range: Self spells, "you" in the spell description means exactly, "you, the spell caster," and never, "you, the spell's target." This interpretation, however, also modifies spells like divine favor, detect evil and good, crusader's mantle (that one's a bit of a pickle to decode with a mount), and every other Range: Self spell. I'd argue it's all or nothing here. Either they all work on the mount (in some way), or none of them do. It doesn't matter how you rule here, but you should be consistent. Given the number and range of Paladin spells that are Range: Self I question an interpretation this narrow as being the design intent, but it's certainly supportable. About the only thing that reinforces this interpretation is the fact that the Smite class ability does not work on a find steed mount, but that's only because the class ability isn't a spell so it doesn't qualify for find steed's expansion.

The difference between Case 2 and Case 3 is deciding if find steed changes the wording of spells to "The first time both you and your mount hit with a melee weapon attack [...]" or changes to "The first time either you or your mount hit with a melee weapon attack [...]". Honestly, there's not enough information to decide either way. The spells are not written with find steed in mind, and find steed is not worded to make the end result clear.

You can argue Case 2 by saying, "The spell is intended to only affect a single melee attack; if it were intended to affect multiple targets, it would be higher level or otherwise deal less damage."

You can argue Case 3 by saying, "Find steed, like find familiar or hunter's mark, is a class ability masquerading as a 2nd level spell, and that wording was put there to have an intended effect. Furthermore, making a Paladin more deadly while mounted -- a fairly rare situation in most campaigns, IMX, and small Paladins are already less threatening -- is in-line with the desired result of the theme and flavor of the class. Given also the relative scarcity of spell slots, the additional power is probably not significant in most cases." This is not a particularly crunchy argument, but given that 5e does not separate crunch and fluff, it is legitimate.

If I were to rule conservatively, I would probably rule Case 2. If I were in a more liberal frame of mind, Case 3 would be reasonable. As it stands, I don't see any compelling justification for any one interpretation.