Ok, this took more time than I had anticipated.  Hopefully this helps.  If not, let's invite some experts to a Friday meeting.  

Here's how I think about the terms:

1.  Housing is called Affordable Housing when the occupant pays no more than 30% of his/her income for gross housing costs including utilities.  This is from HUD - the US Department of Housing and Urban Development and dates to 1981 and the ravages of the Reagan Budget.  Most renters and many homeowners in fact pay more than 30% of their income for housing.  Frankly, I think for people with incomes from 0-30%, this is almost meaningless.  The key is to have enough money left, after you pay your rent, to live on.  I don't consider food, transportation, personal hygiene, laundry detergent/costs, over the counter medications, a pair of shoes that fit - none of those should be in the "disposable" income category - these are the things everyone needs just to live with some basic decency.  My experience is that people with more than enough disposable income, even people of good will, don't get it - they assume that everyone can do what they do and just write a bigger check or pay a little more.  Or one of my personal favorites - "the poor should manage their money better."  Again, my experience is that the poor manage their money incredibly well - it's more money they need - not better management.  

2.  People are called Low Income when their household income is below the median income for their geographic area - some programs use the State Median Income and others use specific geographic areas - HUD, for instance uses Tacoma/Pierce County as one specific geographic area in certain of its programs.

HUD has three categories of Low Income people and various housing programs have specific eligibility requirements.  This goes into the "nothing is easy" file. 

3.  Area Median Income (AMI) for Tacoma/Pierce County (half the people above and half below the income point) 2018 attached below.  Again from HUD.  Many of these incomes are actually pretty decent IF you have a large enough household and IF your income is constant.  But many of these households require two earners to reach their income level - we've all seen it - something happens to one income earner and the household can no longer pay the rent or mortgage.  Another problem is when policy makers, funders, and even supporters make assumptions that everyone within an income class will be at the TOP of the class.  That is, a very low income household is always right at 50% AMI and never at 31% AMI.  So they write regulations assuming the Very Low Income participants will all be at 50%.  

4.  Low Income Housing - I don't think housing itself can actually be "low income"  even though we use that term all the time.  Studs, windows, flooring, roofing, land costs  - their costs generally go up for high income finishes and occupants but every housing unit has certain basic costs to build.  I think we use "low income housing" to refer to the people who will live in the housing - that somehow the housing itself is less expensive and less desirable.  Seems to me that housing ought to look like, well, decent normal housing.  Something any of us would like to call home.  Some programs require that all the housing units are available only to low income households; others may set-aside a percentage or a number of units for low income households.  This also goes into the "nothing is easy" file.  

5.  Attainable Housing - Pierce County Planning and Public Works is using this term.  I don't like it - I find it aspirational.  If we just do the right things, we can attain certain housing.  I also think it complicates the existing discussion, program eligibility, funding regulations.  That's me - you make your own decisions.  Here's the thing - if we keep asking policy makers and funders to quantify the incomes in dollars of the people who will live in the housing under consideration, I think we will begin to change their understanding of the income realities of the people we are trying to house.