Putting the cheat sheet in the first post, reasoning and clarifications will go in future posts.

Standard Offer (aka optional 23)
Formula: offer = board value
Use These: All the time (duh)

No-23 Offer
Formula: offer = board value + 23 value
(or) board value = offer - 23 value
Use These: When the player would normally want to take 23
(ie this offer must always be larger than the standard offer)

Forced-23 Offer
Formula: offer = board value - 23 value
(or) board value = offer + 23 value
Use These: When the player would normally want to avoid 23
(ie this offer must always be larger than the standard offer)

Combination Offer (aka cash+prize offer)
Treat the cash component as forced-23 and the prize component as no-23, then sum the two together. The total should be (roughly) equal to the board value.
Use these: These offers are more flexible, but the typical use is when the player would normally want to avoid 23 at their stake level but would be more interested in it at a lower stake level.

Tandem Offer (where offer 1 is no-23, offer 2 is forced-23)
The larger offer should be calculated as normal, but the smaller offer should actually be calculated according to the standard offer.
Use these: Whenever you are using either no-23 or forced-23, and want to make the choice between them clearer.

Modified 23 Offer (eg no-upside 23 / no-downside 23)
Formula: offer + modified 23 value = no-23 offer
(or) offer = no-23 offer - modified 23 value
(or) board value = offer + modified 23 value - standard 23 value (if positive)
(For this formula, use the standard offer instead of the no-23 offer if it would be larger)
Use these: At anytime, with caution.

First of all, there are no formulas attached to calculating "board value" and "23 value" here - figuring out how to do that is your job! This is simply a guide on how to use (and value) conditional offers appropriately.

So the most important thing going on here is that, for basic no-23 and forced-23 offers, the conditional offer should never be smaller than the standard offer. If you do make a smaller offer, you are effectively offering less than the board value for the "right" to make the choice they would have made anyway. As going to the end and taking your box always gives you the option for which 23 you like, making reduced offers for their preferred 23 option is only going to end in NO DEALs and frustration.
You can also think of it as that the standard offer is giving the player the most flexibility, so removing that flexibility should result in larger offers as compensation.

Tandem offers are essentially "standard offer + conditional offer" with the standard offer wearing fancy dress. The appeal of these is that they make it more obvious as to what is going on, as if you were going to take the 23 choice the conditional offer gives you then there'd be no reason to take the smaller standard offer anyway.

Modified 23 offers can be used at any time, but they are quite a bit more complex to "get right" as several factors must be considered. The most important thing here is that if the player would have wanted to take the base 23 already then its value must be factored in, as the modified 23 offer removes that option from them.