I happen to believe that Scripture states only one baptism of the Holy Spirit, and that is what is often referred to as "POC", or point of conversion...when one first believes the Gospel ( Ephesians 1:13 ). Each subsequent moving of the Holy Spirit is being filled with Him...not another "baptism".
First, let us remember that Paul was writing to "the faithful" believers in Ephesus,
and NOT to the unfaithful ones! ... Faithful believers are the obedient ones.
IMO, your Eph 1:13 refers to receiving the Holy Spirit INSIDE,
which is a sign to the spirit world ...
(1) that you have been positionally sanctified
(2) that the Holy Spirit will be taking you through the process of sanctification
I don't know exactly which baptism Paul was referring to in Eph. 4:5,
but Hebrews 6:2 talks about "baptisms".
I see at least 4 different baptisms ...
(1) -- Baptism into the church (salvation) --- the Holy Spirit is the baptizer
“For by one Spirit we were all baptized into one body (the church)
… and have all been made to drink into one Spirit.” (1 Corinthians 12:13)
The Holy Spirit takes up residence
INSIDE the believer … John 14:16-23; Romans 8:11;
1 Corinthians 3:16; Galatians 4:6; 2 Timothy 1:14; 1 John 3:24; 1 John 4:12-16.
(2) -- Baptism in water (symbolic demonstration of faith) --- man is the baptizer
“Or do you not know that as many of us as were baptized into Christ Jesus were
baptized into His death? Therefore we were buried with Him through baptism
into death, that just as Christ was raised from the dead by the glory of the Father,
even so we also should walk in newness of life.” (Romans 6:3-4)
(3) -- Baptism with the Holy Spirit (mostly power for ministry) --- Jesus is the baptizer
John the Baptist proclaimed that Jesus is the One who baptizes with the Holy Spirit
… Matthew 3:11, Mark 1:8, Luke 3:16, John 1:33.
The "baptism with the Holy Spirit" can come after being born again (or at the same time).
7 verses say that for this baptism, the Holy Spirit comes
UPON the believer, for example:
“… the Holy Spirit fell upon them, as upon us at the beginning. Then I remembered
the word of the Lord, how He said, ‘John indeed baptized with water, but you shall
be baptized with the Holy Spirit.’ ” (Acts 11:15-16)
(4) -- Baptism with fire (extreme suffering for the gospel’s sake) --- Jesus is the baptizer
John the Baptist proclaimed that Jesus is the One who baptizes with fire.
“He will baptize you with the Holy Spirit and fire.” (Matthew 3:11, Luke 3:16)
Jesus asks His 12 disciples if they are able and willing to be martyrs after Him …
“Are you able to drink the cup that I am about to drink, and be baptized with the
baptism that I am baptized with? … You will indeed drink My cup, and be baptized
with the baptism that I am baptized with …” (Matthew 20:22-23, Mark 10:38-39)
All of the baptisms are glorious blessings, even the baptism with fire:
“Blessed are those who are persecuted for righteousness sake” (Matthew 5:10).