Replacing 3-Way Light Switch with 4 wires
Identify the travelers

First, the partner switch is the power source.  You can tell because its common terminal is hooked directly off a dual receptacle.  Unless the receptacle is switched, that must be the supply.  The other two screws on the supply 3-way are the travelers and there is a black and red wire on them.   That is also consistent with the lamps being at the far end; since neutral would be present then, neutral must be white.  So that all jibes. 

Your testing reveals that when you throw the switch, B1 and R wires alternate which one is hot.  So confirmed.

I am a big fan of marking wires with tape to indicate their function. I prefer to use yellow wire for both travelers, since it isn't useful to distinguish them from each other.  I would mark both travelers at both ends thusly. 

There would be no use for any other wires to land on a traveler terminal.  The only remaining terminal is the "common", and assuming this is a 3-way, the remote common is the switched-hot wire going to the lamp.  

I see a /3 cable (containing the travelers and a white) and a /2 cable on the left side of the switch box.  

Loads

Now, in the back of that box I see plain separation between the wires to the left switch, and the wires to the right switch.  With one exception: one neutral crosses over from this group to the other side. That shouldn't happen:  currents should always be equal in each cabLe, and that means a second wire would need to cross over too.  The house may be miswired.  

Obviously B2 goes to the lamp.  But its partner white should be going to the white from the other 3-way, and that's not the case.  Now it would be proper if: 


the lamp was powered from the other box: in that the white wire from the /3 cable will tie to the common terminal, and carry switched-hot back to the other box, where it's tied to a cable to a lamp.  Or
The /2 cable in this box went on to power a lamp, in which case the black wire from the /2 cable goes to common, and its white wire ties into the neutral brought over in the /3 cable.  (because of the equal currents rule, the neutral should not be poached from the neutral bundle in the right side of the box). 


My working theory is that both of the above are true.  W carries switched-hot back to the other box in the /3, and the B2 is the hot from the /2 cable going on to a different lamp.  This circuit powers 2 lamps fed from both boxes.  Or did.  Since W is being used as a switched-hot, B2's partner white cannot return to it; instead they bootlegged neutral from the clump of cables serving the switch on the right. This is bad wiring and should be corrected one way or another. 

If you don't see  where this second lamp would be, look for switched receptacles somewhere in the room.