Combat using a grid and touch spells

Question

My understanding of combat on a grid is that (1) moving onto a new square needs 5 feet of movement and (2) the distance between two squares is 5 feet. A medium creature occupies 5 square feet of space. A normal melee attack's range is 5 feet.
For a medium size creature to attack another medium creature, both of them need to be in adjacent squares. No movement is required if they begin in adjacent squares.

But what happens when you are trying to use a touch spell?

1. Do you just need to be in an adjacent square?
2. Is any movement required?

From my understanding you can't occupy the same space as another creature.

Do touch spells on a grid mechanically function like 5 foot range spells or like something else?

Answer 1

Yes, touch spells work in adjacent squares

If you are adjacent to another creature, you are 5' away, as NautArch explains here.

In the PHB (pg. 202) under Range, it says this about Touch range:

Some spells can target only a creature (including you) that you can touch.

This doesn't really help much, but we can infer that to perform a touch spell on that creature, you do not have to occupy their space, since you can't end your movement there, so touch spells must effectively have a range of 5ft if they are to make any sense (although they don't actually have a range of 5ft or any feet; it's simply useful to think of it as such for the purposes of a grid).

If I include the quote from linksassin's answer from the PHB (pg. 195), spells with a range of touch that require an attack roll of any kind are melee spell attacks (i.e. the spell's description will say "make a melee spell attack") and are therefore subject to this quote:

Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them when making a melee attack. Certain creatures (typically those larger than Medium) have melee attacks with a greater reach than 5 feet, as noted in their descriptions.

Although not all touch spells require attack rolls (since not all of them are attacks, such as cure wounds), we can at least infer from those that do that touch spells can work in adjacent squares nonetheless.

So long as you can touch them without moving out of your 5ft square (i.e. if they are adjacent to you), you can target the creature with a spell with a range of Touch.

Answer 2

Yes touch spells usually have 5ft range, but sometimes they are not mechanically the same

Typically Medium sized creature have a reach of 5ft and can perform a touch attack on targets within 5ft (or on adjacent square of the grid).

Most creatures have a 5-foot reach and can thus attack targets within 5 feet of them when making a melee attack. Certain creatures (typically those larger than Medium) have melee attacks with a greater reach than 5 feet, as noted in their descriptions. PH pg195

The key difference between range Touch and range 5 ft is that touch spells that require an attack roll are usually melee spell attacks. Whereas 5ft range spells are typically ranged spell attacks (though there are exceptions, e.g. Thorn Whip) and are subject to the ranged attack disadvantage in melee.

Ranged Attacks in Close Combat

Aiming a ranged attack is more difficult when a foe is next to you. When you make a ranged atlack with a weapon, a spell, or some other means, you have disadvantage on the attack roll if you are within 5 feet of a hostil e creature who can see you and who isn't incapacitated. PH pg195

Though this of course doesn't apply to spell that don't have an attack roll. So often makes no mechanical difference.

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It may also be worth noting that Touch attacks may be a holdover from previous editions where they were resolved against a different AC