The third thing Sirlin did was make 5 new armies in addition to the classic one, in the vein of Betza's Chess with Different Armies.

This is the part I thought was super cool, as there are some very different armies contained, completely unlike Betza's. I had thought I could just port the armies over against Betza's, but that doesn't completely work, as Sirlin balanced his with the midline invasion and dueling rules.  In addition, the tremendous amount of play they got on the steam game before it was largely abandoned proved a few other holes.  The Reaper army has difficulty winning against Nemisis or Two Kings, but can feel overpowering sometimes against other armies.  Sirlin commented Reaper was very difficult to balance, and it appears it still isn't quite right.

Quoting the rules:

The Six Armies

I) Classic (balanced)
It's regular old Chess. This is the only army that can castle.

II) Nemesis (favors pawns)
The queen is replaced with a new piece: the nemesis. The nemesis piece moves as a queen, but cannot capture or be captured except by the enemy king. (It can check and checkmate a king, and a king can capture it.)

Your pawns can move as normal pawns, or alternatively they can make a nemesis move, which is a move one space toward the enemy king. (Imagine a box drawn around your pawn and the enemy king; moving inside that box is a nemesis move. That move can be toward your back row if the enemy king is behind your pawn). Nemesis pawns can only capture pieces (or threaten a king) the same way normal pawns can: diagonally forward. Your pawns cannot move two spaces at the start of the game.

Despite what the rules say, Nemisis is not really about the pawns.  This army is focused on the Nemisis.  You can run the Nemisis right into the thick of things, unworried about loosing it.  It's quite a good army.  The ability to move your pawns out of the way of your rooks to develop them early also changes things in interesting ways.

Betza calls the whole class of pieces that cannot capture except the king nemisis pieces, from the first piece of that type he came across. 

I'm not sure how easy this is to integrate with Betza's Chess with Different Armies. It might be close.

III) Empowered (favors knights/bishops/rooks)
While a knight, bishop, or rook is adjacent (diagonals do not count) to another knight, bishop, or rook on your team, each piece gains the movement powers of its neighbor in addition to its regular movement powers. (King, queen, and pawns cannot gain movement properties.) To compensate for this power, the queen can only move as a king. Example: if knight, bishop, rook are in a line, adjacent to one another, then knight can move as knight+bishop. Bishop can move as knight+bishop+rook. Rook can move as rook +bishop. The knight does NOT gain rook movement in this example, nor does the rook gain knight movement.

Sirlin doesn't acknowledge any influences for Chess2, but this is almost exactly like one of Betza's ideas.  I can't find the page right now, but Betza had the powers relayed when pieces were within the movement range.  (A rook that was a knight's move away from a friendly knight would also be able to move as a knight.)  This becomes especially apparent when you read the forums on the Sirlin games website--David Sirlin posts there he had to reduce their empowerment range to only orthogonally adjacent pieces; playtest proved that anything more was too powerful.

In my opinion, this army could be made a standard part of Betza's Chess with Different Armies.  It is not too much affected by the midline invasion or the dueling rules, and it has gone through a considrable amount of playtesting and balancing. 

IV) Reaper (favors queen)

The queen is called a reaper. It can teleport and capture anywhere on the board except the enemy's back row. The reaper cannot capture a king.

Also, the rooks are ghosts that can teleport to any open square on the board. The ghosts cannot capture or be captured.

The Reaper army is really awesome—it plays very very different than anything else. Unfortunately, it is balanced with midline invasion and dueling. Dueling keeps the Reaper from running amok....the player needs to worry about what he takes. And the Reaper army really needs midline invasion to win-- they only have bishops and knights to create checkmate.

Betza talks about teleporting pieces lower down in the same article about the nemisis right here.

If midline invasion was allowed for only this team, it might be transportable to Betza's Chess with Different Armies without too much modification.

V) Two Kings (favors kings)

You have no queen, but instead have two kings called warrior kings. If either one is checkmated, you lose. To win by the Midline Invasion method, BOTH warrior kings must cross the midline of the board into enemy territory.

A warrior king can move and capture the same way as a regular king, though it also has the option of doing a Whirlwind attack. For this, the warrior king stays in place and destroys all adjacent pieces—friendly and enemy—including diagonally adjacent pieces. You cannot Whirlwind if your other warrior king is adjacent.

After each of your turns, you may (optionally) take a special king-turn where you only move a warrior king. On your normal turn, there are no special restrictions. You can move either warrior king, or some other piece, whatever you want. During your king-turn, you may ONLY move a warrior king or perform Whirlwind with a warrior king. It doesn’t matter if you moved that warrior king or not during your normal turn.

You can’t move into check on your normal-turn or your king-turn.

Helpful hint: whenever you choose to skip this extra king-turn, it would be helpful if you tap one of your warrior kings as a signal to your opponent that he can take his turn.

Two Kings is also very different. The whirlwind attack is very powerful, but it requires you moving your king/s out in front. The delicate balance between attacking with and keeping your kings alive is a very different and fun experience!

I don't recall hearing anything like this in any of Betza's articles.

I'm not sure how much Two Kings relies on the midline invasion rule to be balanced. I don't think it matters much and can be directly ported over to Betza's Chess with Different Armies. But I might be very wrong on that!

VI) Animals (wild card)

Knight -> Wild Horse. Moves as a knight, but can capture its own pieces.

Bishop -> Tiger. Can only move up to 2 squares diagonally, but does not move when it captures (immediately jumps back to the square it attacked from).

Rook -> Elephant. Can only move up to 3 squares orthogonally. It can capture both friendly and enemy pieces, even multiple pieces in one move. If it captures a piece, the elephant rampages and must move its maximum distance, capturing everything in its path. Also, the elephant cannot be captured by a piece more than 2 squares away. (Draw a 5x5 box with Elephant in the center. It can't be captured by pieces outside the box.)

Queen -> Jungle Queen. Can move as a rook or as a knight.

This is a really delightful army, it takes a little bit to develop but is powerful. I really like both the Elephant and the Tiger—they push the envelope differently than Betza's pieces.

In my opinion, this can be moved over to Betza's Chess with Different Armies as-is. The pieces are about as strong as their FIDE counterparts one-on-one, so it should slide seamlessly in.

And I have a few more quotes from the official rulebook on miscellaneous stuff:

Choosing Your Army
Players choose their armies in a simultaneous, double-blind fashion at the start of each match. It’s permitted for both players to choose the same army. Though players will likely specialize in playing only one army, in a multiple-game match, the loser of a game may switch to any army for the next game. The winner of the previous game may not switch.

Promoting Pawns
When one of your pawns reaches the last row, you must promote it (not optional). You can promote to any piece that’s part of your army other than a pawn or a king (or a Warrior King). For example, a pawn on the Animals team could promote to a Tiger piece, but an Empowered pawn can’t promote to a Tiger because Tiger is not part of its army. When you promote a pawn, your opponent does not get a stone.

Dueling Ranks
For purposes determining if you have to pay 1 stone to initiate a duel against a higher ranked piece, the only possible ranks are 1) pawn, 2) knight/bishop, 3) rook, and 4) queen. In other words, all special queens count as queens, even though the Empowered queen is rather weak. Elephants count as rooks. The wild horse and the tiger count as a knight/bishop. Nemesis pawns count as pawns.

Draws
There are no stalemates in Chess 2. The other types of draws from Chess 1 still apply here, though they are much more unlikely because of the Midline Invasion rule. The types of draws are: threefold repetition (when the same position occurs three times), the fifty-move rule (when the last fifty successive moves made by both players contain no capture or pawn move), and impossible checkmate (when neither player has sufficient material to checkmate, and Midline Invasion is not possible).

Other Notes

All pawns on all teams have the ability to en passant.

Pieces cannot pass through the Reaper army’s ghost rooks or occupy the same square as a ghost rook.

A warrior king’s Whirlwind cannot destroy a ghost rook.

Even the reaper cannot take an elephant if its more than 2 squares away