When you have 2 way switching (Which is what you would like) you require 3 cores interlinking both switches. Your electrician has installed such a cable (The one with grey, brown and black) When you add a 2nd light to the first, you should connect this new light directly to the first existing light, so that either switch will operate both lights at the same time. In your case this has not been done.
In order to progress further the cables would need to be tested to establish what they are connected to, and since anyone who reads this is only sitting looking at a screen and not actually on site, it is not possible to tell you which wire goes where, you are going to require a competent person to visit site and establish what does what and connect it accordingly.
I can not help but wonder does your electrician know what he is doing.
I am a little uncertain as to what you have or want. So to start with two way wiring. In a standard single switch the line in and line out go to com and L1 these names can vary with com some times called L but it is normal to find com other side of switch block to L1 and L2 (if in a one way switch L2 exists).
However with two way wiring theses two wires go to L1 and L2 so first thing is move the wire from com to L2.
Now the three core, does not matter which colour you use but either end is wired the same, so if brown goes to com in one switch it also goes to com at the other switch.
So you end up with original switch two wires in L1 and L2 and one wire in com, and new switch one wire in each. That is standard two way wiring.
So moving on, electronic switchs will has an S terminal instead of com and in the main wired the same way. You can also get wifi switches which remove the need for wires between the switches, some switches need batteries to work remote, some use the energy from switching there is no need to actually run wires between the switches, however the wired method is the cheapest.
Now you say two outside lights, this is where it gets complex, if the second light is from the same circuit you could put the line feed into a chock bloc and not use it, and take a supply from L1 or L2 which ever is switched, if you select wrong one light would not switch off, so try and if will not switch off swap to other one.
However if the lamp is from another circuit then you can't combine, as it would cause what is called a borrowed neutral, actually you are borrowing the line, but called borrowed neutral.
Your post is not good enough for me to work out what you have, if your not sure play safe and get an electrician.
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