I have been looking for the texture that's on these two pictures everywhere and I cannot find anything similar.
Generically, It's also called a moiré effect or pattern. It is not limited to printing halftones but can be produced with any repeating regular line or dot pattern interfering with (overlaid onto) itself.
While some may erroneously refer to it as a halftone or a screen pattern, the appearance of "circles" or rosettes are caused by the interference pattern of more than one layer of the screen (line, dot, or variation). That's the subtle difference. A halftone pattern does not have rosettes while an Amplitude Modulated halftone screen process using four-screens in register at screen angles does.
Here is a (very coarse/greatly enlarged) halftone screen example (shown at 45° screen angle) of a flat tone:
Here is a moire effect of several screens causing rosettes:
Here is a different kind of rosette pattern (open)
There are two different kinds of rosettes depending whether the centres are open or closed. As shown in your illustrations in the original question, both are mis-registered. Originally, maybe they were intended to be open. Different screening practices exist in Europe and in the US.
To answer your question, the effect you show cannot be made using a simple halftone; but, requires the interaction (overlay) of several halftone screens at different angles to create the rosette (a kind of "moire") pattern you display.
This only refers to Amplitude Modulation halftone screening. Frequency Modulated (FM) or stochastic halftone screening has no regular pattern, no rosettes, no screen angles and has no characteristic pattern or texture.
Both AM and FM create a halftone pattern although only the AM screens interaction can create a rosette as shown.
Thanks to bogl for encouraging me to make a better submission. Better late than never. More can be searched using the terms I used in my edited answer.
This is actually not a texture, but a halftone pattern. Many graphics applications are able to render images with halftone patterns for printers with few colors (e.g. 1 or 4).
Related question at Graphic Design SE: How make this halftone round texture?