I have heard lines such as Teeny lines and other similar lines referred to generically as integrated lines for at least 10 years.

The 'integrated' comes from the construction, which essentially marries a shooting head permanently to a shooting line.

It does seem that every company has specific names for their own versions - but probably because several companies, including Teeny, S.A, and Rio have more than one distinct version of integrated line. Examples in the S.A. line are the Streamer Express, Wet Tip Express, and new Custom Express integrated lines which all have different characteristics.

And it is not exactly the case that a heavier line will sink faster. If this were so a 12wt floating line would sink faster than a 200 grain integrated line. Actually the density of the line dictates sink rate. Given two lines of equal density, the heavier tends to sink faster.