OSPF LSA Types Explained

R1#show ip ospf database
Link ID         ADV Router      Age         Seq#       Checksum Link count
10.0.13.1       10.0.13.1       1865        0x80000007 0x002244 3
R1#show ip ospf database router 10.0.13.1

   Link connected to: a Transit Network
     (Link ID) Designated Router address: 10.0.13.3
     (Link Data) Router Interface address: 10.0.13.1
      Number of MTID metrics: 0
       TOS 0 Metrics: 1

I am a little confused with this Link ID for type 1 LSA, when I type show ip ospf database LInk ID for that LSA T1 is 10.0.13.1 , but when I am looking into: show ip ospf database router 10.0.13.1 for that IP it shows me the DR link ID : (Link ID) Designated Router address: 10.0.13.3, shouldn’t be the same?
I feel like there is something that I don’t get, can you help me? Thanks.