Those are not threads, just the pattern formed by the screen. Try looking at any colour printing in a book. With multicoloured printing there is a screen for each colour usually 4 colours but some expensive processes use more. Each screen is arranged on a grid and the angles of the grid are carefully selected to avoid a moiré effects and to limit the number of dots from each screen overlaying each others. Most modern 4 colour print when magnified appears to be arranged in circles. Other systems use transparent or blending inks and the dots are less obvious. With Silk Screen the dots appear to be more randomly distributed due to the differences in registration - Fashion Designer scarves can be created with over 20 screens, murderously difficult to get good results. Spot colour printing will use a screen that is too fine to be seen but sufficent to stop the ink spreading, it appears to be solid colour. The finest art printing is done in a waterless process using screen of 2,540 dots per inch, this gives near to photographic results.
Ink jet printing and electrostatic printing are changing all that, inkjets can use as many colours as can be needed and for glass printing use use UV curing inks that create a near permanent result with heat. This method will replace lithographic printing on glass in the near future. Apart from a massive reduction in energy costs, it has the advantage that short runs are feasible and every impression can be different. It can of course be used for printing UV cured transfers too. Electrostatic printing uses low-heat curing plastic inks and its only application in glass that I know of is for large volume containers as an alternative to litho printing - I don't think it is very common..
There were a lot of transfer printers but some companies bought their own presses to save costs. The inks are finely crushed glass and a carrier printed on transfer paper, heat fuses this into the glass. Transfers could also be painted by hand.