Find steed is intended to give different options than find greater steed

First, some unofficial guidance per Jeremy Crawford, lead rules designer:

Q: The Find Greater Steed kind of bugs me because 5E has so far explicitly avoided 'greater' spells and allowed people to beef up spells by spending larger spell slots. So why is Find Greater Steed it's own spell?

A: Similar to the "greater" spells in the Player's Handbook—greater invisibility and greater restoration—find greater steed provides options that its lesser counterpart doesn't. (emphasis mine)

So, the designer states that FGS provides access to options that FS does not allow. This seems to imply that, though unwritten, there was the assumption that the DM adjudicating FS would not allow the player to summon creatures much more powerful than the ones provided already in the list (which range from CR 1/8 to CR 1/4). Otherwise there would be no restriction on the steeds since "Your GM might allow other animals to be summoned as steeds." technically enables anything.

Previously it may have made sense to allow stronger steeds using find steed

Before Xanather's Guide to Everything was published, it makes a lot of sense to use the DM leeway built into the spell to summon more powerful creatures in DM-allowed cases. For example, a 20th level paladin seems powerful enough to be able to cast a spell giving them something other than mundane animals.

Specifically, because there were no other options available to summon more powerful steeds, DMs had to resort to using the spell in this way because there was no other way for a paladin to summon a steed higher than CR 1/2.

Even Crawford explicitly said that the DM could choose what was summoned (before XGtE):

Find steed lets you summon a warhorse, a pony, a camel, an elk, or a mastiff. Anything else is up to the DM. #DnD

Find greater steed should now be the only way to summon more powerful steeds

As, time goes on, D&D is always changing. Rules get added and changed. Errata gets published, holes get patched, etc. In this case, a new spell was added: find greater steed. This spell was clearly added with the intent to be a more powerful option than find steed:

1. It is a 4th level spell (vs find steed's 2nd level)
2. It is called "greater" explicitly saying that it is intended to be more powerful
3. In it, all the listed steeds are much more powerful than those listed in find steed

The very existence of find greater steed means that there was a limitation to begin with on find steed or that one was intended. What reason would the designers create a spell that would essentially replicate the effects of a lower level spell with no benefit? Why create a "greater" version if it can already be done with the original version? The answer: this is because the intent is for find steed to only be used with less powerful steeds while find greater steed would work for more powerful ones.

Another reason supporting this is that RAW imply that find steed can summon any steed with a 2nd level spell only. However, since the designers clearly have rated summoning pegasi and pertyon (eg) as a 4th level spell-worthy ability, it seems that continuing to allow this would mean that you are severely bending the spell slot economy for this spell only. Thus, making find steed much more powerful than comparable 2nd level spells. This is generally not seen as a good thing, and it would seem not to be in the interest or intent of the designers of the game to want/allow this.

What to do if a DM has already allowed more powerful steeds with find steed?

DM should do the same thing that happens when any new rule is added or old rule changed or patched: either accept the new going forward or ignore it and keep doing the way you were doing before.

Regardless of how DMs previously ruled, going forward we now have an official option that enables doing the things DMs previously had to adjudicate using the lower level spell. This provides the advantage of some additional clarity and allows for the game to be played according to the designer's intent.

If intent isn't important or if it would create an issue for the DM or players, the DM could, of course, just ignore the new spell and keep adjudicating with whatever houserules they previously used.

tl;dr

Rules as intended suggest that only FGS be used for more powerful steeds and FS used only for weaker ones.

but

RAW would technically still allow the DM to choose a mount covered in FGS with FS.

You quote this bit:

Your summon a steed that takes on a form that you choose: griffon, pegasus, peryton, dire wolf, rhinoceros, or saber-toothed tiger. The steed has the statistics of the chosen form

and by that, the answer is very straightforward. Size is one of the statistics of a monster. In fact, it's the first listed in the "Statistics" section in the intro to the Monster Manual. Since the peryton has a size statistic of medium in its description, that's the size of the summoned steed when you choose that form.

Note that non-greater, regular Find Steed also has medium-sized options (pony and mastiff). Presumably these sizes are there to support small-sized paladins (gnomes, halflings, etc.).