I know this one is quite old, but after a few recent papers on Arxiv about delicate primes, I added the simple function to my library.  Some searching afterwards found this puzzle.

In celebration of 2021:

#  500 digits: 2021 * 10^(500-4) + 7543997
# 1000 digits: 2021 * 10^(1000-4) + 2550219
# 2021 digits: 2021 * 10^(2021-4) + 4523733

Found using Perl/ntheory.

Double checked prime with Pari 2.14 ECPP, and double checked as a delicate/weakly prime using Michel Marcus' Pari function.

I believe they are the same (technically "digitally delicate primes" to reference base 10).  OEIS A050249, "Weakly prime numbers … Also called digitally delicate primes."

The following is just trivia because it was interesting for me to search.  There is no consistent answer.

Klamkin's 1978 question doesn't seem to call them anything, nor does Tao's paper.
I'm not able to track down Erdos' 1979 "Solution to Problem 1029" to see what he says.
Mathworld and Anderson's Wikipedia article calls them Weakly Primes, referencing your page.
Hopper and Pollack's 2015 arxiv paper calls them "digitally delicate":

     "The infinitude of the primes appearing in Corollary 1.2 had earlier been shown by Erd˝os [2].
      (He assumes a = 10 but the argument generalizes in an obvious way.)
     When a = 10, these “digitally delicate” primes are tabulated as sequence A050249 in the OEIS, where they are called “weakly prime”.

The papers by Filaseta and Southwick, most in the last decade, also call them digitally delicate.

Emily Stamm's 2020 preprint calls them weakly prime, and references most of the other papers so clearly the author is familiar with both terms and chose that one.