The merchant record with the photo looks like a good match, I saw this a while ago, but need more evidence this is him. Joseph and his brothers were all born in Birkenhead but they all moved to Liverpool.
His Merchant Navy CR10, as outlined by KGarrard also gives the name/s or official number of the vessels he sailed on. If we take the first ship for 1919, LEITRIM official number 139108 Her crew agreement and her logbooks for 1919 is available from here,
https://www.mun.ca/mha/holdings/viewcombinedcrews.php?Official_No=139108The crew agreement, should contain details of his last address, Next of Kin and maybe other personal details which may give you proof of his identity.
They charge around £25 for this service. Just make sure you ask from the crew agreement covering the date 6/Jan/1919 and any details of him in the narrative pages of the logbook. Also quote the Discharge A number 451879.
I would not think it unusual that seamen born in Birkenhead gave there place of birth as Liverpool.
If it turns out it was your man. You can find out the other vessels he sailed on via the "Clip" website.
http://www.crewlist.org.uk/data/vesselsnum.php
