Can a Battle Smith use Replicate Magical Item to grant their Steel Defender Plate Armor

To infuse an item, you need the nonmagical version of that item.

The artificer's Infuse Item feature states (TCoE p. 12, E:RftLW p. 57, WGtE p. 179; emphasis mine):

At 2nd level, you gain the ability to imbue mundane items with certain magical infusions. [...] Whenever you finish a long rest, you can touch a non-magical object and imbue it with one of your artificer infusions, turning it into a magic item. An infusion works on only certain kinds of objects, as specified in the infusion’s description.

The Artificer Infusions section at the end of the class description states (TCoE p. 20, E:RftLW p. 61, WGtE p. 181; emphasis mine):

The description of each of the following infusions details the type of item that can receive it, along with whether the resulting magic item requires attunement.

And finally, the description of the Replicate Magic Item infusion states (TCoE p. 22, E:RftLW p. 63, WGtE p. 182; emphasis mine):

[...] Alternatively, you can choose the magic item from among the common magic items in the game, not including potions or scrolls.

In the tables, an item’s entry tells you whether the item requires attunement. See the item’s description in the Dungeon Master’s Guide for more information about it, including the type of object required for its making.

Each of these portions I've bolded make it clear that you need a nonmagical version of the item to infuse, whether you're using Replicate Magic Item or one of the other artificer infusions.

Even though the Dungeon Master's Guide doesn't actually say "[X] is the type of item needed to make [Y]" for any listed magic item (as far as I know), this sentence seems to refer to the fact that each magic item that can be replicated is essentially an enhanced version of a clearly identifiable mundane item.

For instance, boots of elvenkind are boots, so the clear implication is that you would need mundane boots to infuse if you want to use Replicable Magic Item to make boots of elvenkind. This follows the same logic as the Boots of the Winding Path infusion (TCoE p. 21, E:RftLW p. 62, WGtE p. 181), which does explicitly state:

Item: A pair of boots [...]

In the case of your example, the description of cast-off armor states (XGtE, p. 136; emphasis mine):

Armor (light, medium, or heavy), common

Thus, you need a nonmagical version of whatever kind of armor you want to turn into cast-off armor. If you want cast-off half plate armor, you need a nonmagical set of half plate armor to imbue with that infusion.

The Replicate Magic Item infusion doesn't create a magic item from thin air - it gives you the ability to imbue a certain mundane version of that item with the necessary magic to function as that magic item.

Yes, if you want

The rule for the Steel Defender states:

You determine the creature's appearance and whether it has two legs or four; your choice has no effect on its game statistics.

The wording is sufficiently vague that you could say "My steel defender has reconfigurable limbs, allowing me to change it between 2 legs and 4."

The rules do not say that you determine the Steel Defender's appearance when you create it. Obviously, this does happen, but the wording permits the interpretation that one can choose to alter its appearance subsequently because no timing is specified.

And why should they? You might also paint it or glue a wig on it. These are cosmetic changes permitted by the fact you're playing an open ended RPG and do not violate any explicit rule.

Since your DM is so "by the book," it's worth remembering that Rule zero is in the book and grants wide leeway to DMs and players to act as they want.

• The players describe what they want to do.

• The DM narrates the results of the adventurers' actions.

Given the fact that we're dealing with a cosmetic or narrative decision, you have plenty of (ahem) leg to stand on.