Here's an example from a printed book of Central Europe, 1839, :
Eastern Europe, 1873:
And Israel, 2002:
When Rav Kook wrote letters to Sefaradim, to show respect and esteem he would use this script (following in the beautiful tradition of Abarbanel, who penned a famous, lengthy letter to an Ashkenazi rabbi in his script). Here's R. Kook's signature:
The following excerpts are from the Encylopedia Judaica (2nd ed.):
MASHAIT SCRIPTThere are things to quibble with in the above, but as a standing description it will do.
Cursive was sometimes used as a book hand when, of course, more care was taken with the writing. But this did not involve an approximation to the square; it led to elaboration and ornamentality of a different kind, and thus to the shaping of a new style, the mashait script (incorrectly designated "rabbinic").
. . .
The usual, but incorrect, designation for it is "Rashi script," obviously because *Rashi's commentaries on the Bible and Talmud – the books which everybody was constantly handling from boyhood to old age – were printed in (Sephardic) mashait. Rashi himself, naturally, wrote in Ẓarphatic (see below).
(Here is the EJ's example of Zarphatic, from 1182:
Why do scholars call the script mashait? They are following the original name that this script was known in days gone by.
The first to inquire as to the meaning of the name is Elijah Levita. The following is from his Sefer Tishby pg 105 (1541, Isny edition):
As you can see, his entry has it as "mashqeit." He was informed that it was Arabic, but after inquiry learned that was not the case. He asked many Jews of different ethnicities; Italians, French, Spanish, Greek and Arab. All used the name, but not could explain it's meaning. He further informs that Ashkenazim call it "masheit," and of course they do not know why either.
As it turns out, there is another place where Levita discusses the word, in a letter written in 1531 to Sebastian Muenster ("איש נבון וחכם אדוני זיבאשטיאנוש מונשטירוש"). Published by Moritz Peritz in MGWJ 1894 pp. 252-267).
His treatment in this letter is as follows:
Here he claimed that משקיט is definitely Arabic and means small and thin, a reference to the alphabet's appearance (relative to the normal square Hebrew, one presumes).
I'm not sure when the Tishby was written, although I know (via C.D. Ginsburg) that Tishby was one of three manuscripts Levita took with him when he moved from Venice to Isny (1540). In any case, I am pretty sure it was written after 1531 (date of the letter to Muenster). It would be nice to know precisely what led him to change his mind about it being Arabic.
In any case, H.J. Zimmels collected the references to the script in many responsa and other sources, and found the following:
Among Sefardim the script is called משקא, משקייט, משקיט, משקי ,מאשק and משק, as well as other names which are not relevant because they are entirely different names.
Among Ashkenazic sources, the following variants are found: משיט, משיטא, משייט, מעשיט, משטיט, גימשיט, מעשוט and משיטה.
Chida cites an early source for the association of this script with Rashi:
Interestingly enough, this source is not Ashkenazic, רדב"ז having been born in Spain in 1479. After the Spanish exile, he spent the rest of his life in Morocco, Egypt and Eretz Yisrael.
So where does that leave us? We still don't know the precise meaning (or I don't). However, I did see a suggestion that it actually does come from the Arabic סקט سقط , meaning some variation of fall, downgrade or to degenerate. Perhaps it referred to the status of this alphabet as the vulgar script. C.f. the reasons given by Maimonides (the cursive enabled people to write profanely, reserving the square Hebrew for sacred texts only) and Rema (the cursive was ideal for writing down Oral Torah, which really should not be written at all, hence the development of an alternative, vulgar script.)
In any case, I see no reason why it can't also be called Rashi script. It might be a quirk of history that it has no true association with Rashi, but what a fitting honor that the Rabbinical script came to be known after him!