In my case, I used PSI-BLAST in the local BLAST+ to search queries against the self created database. After the first search, I added the searched result to the database and then conducted the multiple sequence alignment myself with MUSCLE to get the conserved parts. Then, I searched the queries against the consensus sequence obtained from the multiple sequence alignment (without gap). In this process, I did not calculate a PSSM, while the first iteration of PSI-BLAST uses a constant matrix, which is much similar to BLASTP. So what did I actually use in the process – PSI-BLAST or BLASTP?
You simply used BLASTP against a consensus sequence. It is not how PSI-BLAST works. The first iteration of PSI-BLAST is identical to BLASTP.