Which Smash Ultimate characters have additional moves?

Solution 1:

There are a number of characters that have additional moves, but first, you are missing a few sets of moves that all characters have:

• 1 Final Smash
• 3 grab moves (regular/pivot/dash, which have the same animation but different frame data)
• 1 pummel (press A while you have a character grabbed)
• 4 throws (up, down, forward, back)
• 3 taunts (up, down, side)
• 2 rolls, a dodge, and a shield drop (forward, back, and down while shielding, and very slight down while on a platform)
• 1 air dodge
• 4 options from ledge (normal getup, attack, jump, roll) (previous games had two of each of these, depending on if you were over 100%, not sure if ultimate has this or not)
• 4 getup options from being knocked down (normal getup, attack, roll left, roll right)
• 3 tech options (tech in place, tech left, tech right)

Here are some other additional moves specific characters get under certain circumstances:

• As mentioned by Mage Xy, Zair moves if you have a tether grab (young link, toon link, samus, dark samus, zero suit samus, lucas, luigi)
• As greyShift has mentioned, Ryu and Ken have different ways of inputting their attacks which grant them more power
• Also as greyShift mentioned, they have "medium" attacks in addition to their tilts and smash attacks
• Bowser Jr has an aerial attack after having used his up B
• Donkey Kong has the option of throwing again after using his Cargo Throw (forward throw)
• Robin has different aerial attacks when you smash the control stick rather than tilt it, as with smash attacks vs tilts on the ground. The levin sword must be available to perform these.
• As you mentioned, Inkling has the special option of pressing B while shielding
• Samus and Dark Samus have two different side-Bs, depending on if you smash the control stick or not. Smash side-B produces a super-missile as opposed to a regular missile.

And for completeness, here is a list of the characters whose moves are actually completely separate moves, however they simply replace other moves:

• Kirby has 78(?) different neutral Bs, depending on who he has eaten
• Little Mac has a different neutral B depending on if his KO Punch is charged
• Cloud has a different Down B according to whether or not his Limit Break is charged. Also all his specials are different once Limit Break has been charged.
• Ken and Ryu both have two final smashes, depending on where the opponent is located
• Worth mentioning that technically Pokemon Trainer is one character, so all moves are duplicated three times between Squirtle, Ivysaur and Charizard.
• The Miis each have 4 options for each special

Solution 2:

Ryu and Ken have additional attacks used by inputting their Street Fighter inputs.
Both characters have a stronger Shoryuken(up-B) they can do by pressing forward, down, downforward, then A or B. This does 1.25x damage and has some additional properties.
They also both have the Hadouken (neutral B) input, which is down, downforward, forward, then A or B. Ryu can also do a multi-hitting red fireball with back, downback, down, downforward, forward, then A or B. There is also Tatsumaki (side-B) for both characters using quarter-circle back, and Ken has at least one more input for an overhead kick which I think is quarter-circle down. I'll link a guide later for the full list of special inputs.

They also both have access to strong and light tilts. If you press utilt and release the button quickly, it will be a quick, soft, and spammable elbow that easily confirms into shoryuken. If you press utilt and hold the button, it is a slower strong elbow that has invulnerability on his upper body. All of the tilts and jab have this light/heavy option.

I'd also like to mention that Marth/Lucina/Roy/Chrom can change hits 2,3, and 4 of side-B by inputting up, down, or neutral during each input.