Q:

How to play against the inverted Philidor opening with Black

I was playing with the Black pieces online today, and faced a very strange opening:
[fen ""]
1.  e4 e5 2.  Nf3 Nc6 3.  d3 Nf6 4.  Be2

I check the database, and it seems that 4...d5 is good for black but I didn't understand the strategy behind it at all.  Here are a few moves from the database:
[fen ""]
1.  e4 e5 2.  Nf3 Nc6 3.  d3 Nf6 4.  Be2 d5 5.  Nbd2 Bc5 6.  O-O O-O 7.  c3 a5 8.  Qc2 Re8 9.  b3

I don't understand what Black is playing for here.  Like what the plans are.  I would really appreciate some feedback or references where I can read more about it.  My understanding is that the Black pieces look like a position from an Italian opening.  But I felt that the Italian plans don't work here tactically (like maneuvering Nc6-e7-g6). 
I'm also open to other suggestions, such as playing 5...g6 (but I also didn't understand the plans behind it in this position). 

A:

The basic idea with ...d5, ...Bc5 is just to gain some centre space.  I wouldn't really compare this position's plans to the Italian, as they're not that similar.  Black's pushed ...d5 early on, and the opponent's (White) knight is on d2 instead of c3. 
Analysis for 8.b3: one move for Black in the database is 8...Re8 (or 8...Qe7), supporting the e5-pawn.  Now playing ...d4 is an idea, since cxd4 can be answered with ...Nxd4 (as the e5-pawn won't hang).  In the case of 9.Bb2, 9...Bb6 10.Re1 Bg4 11.Qc2 Qd6 looks nice for Black.  Pushing ...d4 at some point is still in the cards.  Black could also consider some ...Qc5 move, pinning the c3-pawn and targeting the f2-pawn.  Adding more pressure on the d-file with ...Rad8 looks reasonable. 
Analysis for 8.Qc2: 8...Re8 9.b3 could be met with 9...Ba7 (or 9...Bb6), preventing White's queen from x-raying the bishop.  This makes ...d4 safer to play.  After 10.Bb2 h6 (stopping Ng5 if Black decides to play ...Be6 at some point) 11.Rfe1 Qd6 12.Bf1 Bg4, we have a similar position to what was reached for the line looked at in 8.b3.  The plans are similar to what they were there.