NavList:

First, let's consider what will not change, no matter how we clear the sights (assuming the methods are reasonably accurate and we make no errors). The plotted lines of position wil be in the same locations for the same sights, within margins of calculation accuracy, for any assumed position that you choose so long as it isn't too far from your actual observation location. The size of the distance that qualifies as "too far" depends on your requirements, but typically it's 30 nautical miles, sometimes as much as 60 nautical miles. This is something that you should try out: take any one of your Sun sights and use the USNO web app to get Hc and Zn data from two slightly different locations, differing for example by 4' in latitude and 7' in longitude. Then take the resulting Hc, subtract your Ho and plot the resulting intercepts and LOPs. See what happens? The process will yield almost exactly the same line on the chart, despite the fact that you picked different assumed positions. The assumed position doesn't matter, as long as it's reasonably close.

Many modern sight reduction tables require very specific choices for the assumed positions: integral degrees for latitude, and degrees and some specific number of minutes for longitude such that the LHA of Aries works out to integral degrees. This is the specific requirement of short tabular lookup methods like H.O. 249 and H.O. 229 (formerly "H.O." publications). This is the "trick" that permits the tables to be so short.

 The intercept method, by and of itself, has no such requirement, and one can pick any latitude and longitude as the "assumed position". Tables like H.O. 211 (one among many) allow and in fact encourage the use of the best estimated position, typically the DR position, as the assumed position. At the cost of a longer computation, this has a couple of advantages. First, for a navigation enthusiast doing backyard experiments, when we use our actual location as the assumed position, the intercepts can be immediately interpreted as the errors in our sights.

If you get an intercept of 10 nautical miles, then that means your observation was off by 10 minutes of arc (note: this exact linearity holds down to very low altitudes, as low as 5 degrees above the horizon, but it does change very slightly at the lowest altitudes). The second advantage of using the DR as the assumed position is that you do all your plotting from one location, and you do not need to worry about measuring off longitudes to locate the distinct assumed position for each sight that you take. The plot is cleaner and simpler.

Finally, it's worth noting that the line of position is the fundamental output of any analysis of an observed altitude in celestial navigation. There are many ways to get to that LOP via the intercept method, as above, but there are also other approaches. We can calculate individual points on the LOP for any given latitude without even mentioning an assumed position. The line of position doesn't care how we calculate its coordinates. 

I see from a post you made while I was writing this that you already understand what's going on, but I hope there's still some benefit here.