I'm soon to embark on a trip to the Caribbean and Hawaii, travelling BA to San Jose and then AA from there to Hawaii. The trip is doubling as a Tier Point run

I'm however so confused by AA's cabins and fare codes. Some list the sector as 'First' others as 'Business'. Just to explain:

San Jose - JFK (2 class 757). 'Business Class'. Fare code 'I'
JFK-LAX (3 class 767). 'Business Class' Fare code 'I'.
LAX-HNL (2 class 757). 'First Class'. Fare code 'A'.
HNL-DFW (2 class 767). 'First Class'. Fare code 'A'.
DFW-MIA (2 class 737). 'First Class'. Fare code 'P'.
MIA-San Jose (2 class 757). 'Business Class'. Fare code 'I'.

So even those the aircraft i'm flying San Jose to JFK on is identical in configuration as the one i'm flying from LAX-HNL in, even if i'm sat in the same cabin on the very same seat, on one sector it is called Business and the other it is called First. Why is this?

And the big question for me is Tier Points? As far as BA are concerned for calculating TP's are sectors listed in 'First' measured at the same level as 'Business'? When using the 'how many Avios' tool on ba.com on all the sectors i'm booked in 'First' and selected this class from the drop down list is says that class is not available. And yet on ba.com it lists AA fare classes 'F/P/A' as 'eligible First Class codes'.