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The term "cut checks" goes back more than 150 years, but the usage is complicated by the fact that in some instances "cut checks" signified "canceled checks" while in other it signified "created checks." The allied term "cutting checks" goes back at least 147 years.

'Cut checks' as canceled checks

The earliest instance of "cut checks" (in the sense of physically canceled checks) in Google Books search results appears in Fraudulent Canal Scrip: Report of Evidence before the [Illinois State] Senate Committee on Finance (February 4–15, 1859):

Mr. Fuller. Have you a recollection of packing this unfinished scrip in the box at that time?

A[nswer, by Joel Manning]. I can't say that I have.

Mr. Cook. Have you a recollection of seeing it in the canal office subsequent to the packing of the boxes?

A. No, sir. I know we had some left uncut at the time the boxes were packed. Some of it is cut, and I know it is cut the way I cut checks. I have no doubt I packed up all there was, but I cannot remember now.

Mr. Browning. You have no recollection of canceling any of the unfinished checks at the office?

A. No, sir, nor the others either, only I know I did cancel checks, and did it with a chisel—that is about all I know.

From People of the State of New York v. Morris Speigel (December 12, 1892), reprinted in New York Court of Appeals. Records and Briefs (1894):

Q. They gave you a check which had been cancelled, and told you to make a die like the canceller? A. Yes sir.

...

Q. Did you have any idea that this [die] was intended for a stamp to commit fraud, when you manufactured it? A. No, sir.

Q. Who told you what it was for? Did anybody state to you and tell you what it was for? A. I did not know what it was for; I knew it was for cutting a check, or anything else.

Q. It was a mere check canceller? A. Yes, sir I did not know the purport or whatever it was to be used for.

Q. You do not know now that it was to cut checks— A. No, sir.

Q. Is there anything on it to indicate what it is for? A. It is a cutting die.

Q. Might it cut anything of that shape? A. Yes sir.

From Montague Glass, "The Center of Population," in Munsey's Magazine (August 1908):

At this juncture, Mr. Goodel's only clerical assistant, the sixteen-year-old Jimmie Brennan, entered and deposited a bundle of canceled vouchers on his employer's desk.

"Now, Mr. Goodel," he said, “dat guy at de bank wanted me to sign a receipt for dem cut checks.”

“And did you?” Mr. Goodel asked.

“I did not," Jimmie replied, and produced the unsigned receipt from his breast-pocket.

And from Alden Moore, It's Your Business: Your Use of Your Money to Buy Better (1949) [combined snippets]:

When you pay by check, your cancelled check is your receipt that is indisputable proof of payment. These "cut checks" are all returned to you with the bank's periodical statements of status of your checking account. Some banks also make films showing both sides of each check they pay.

Further complicating this sense of "cut checks" is the fact that in a patent application for "Ticket-Boxes" filed on June 11, 1877, the inventor refer to the device as containing "a machine for cutting pieces or checks from tickets," suggesting that "checks" in that instance referred to something similar to "chads"—and not to "bank checks" at all.

'Cut checks' as created checks

The earliest instance of "cut checks" in the sense of "create or make out checks" that a Google Books search finds is from George Jackson, A New Check Journal, upon the Principle of Double Entry, sixth edition (1841):

In Banking Accounts, we should number both the check and tab of the Check Book ; so that there be corresponding numbers on each : and when we spoil a check, write "cancelled" on the tab, and take the next number, it being a had practice to cut checks from the end of the book to supply deficiencies as we proceed. The checks we cut out for temporary purposes should likewise correspond in number to the tab or margin we cut them from.

From United Sates Patent Office, Annual Report of the Commissioner of Patents for the Year 1870, volume 2 (1872):

103,146. — HAND-STAMP FOR CUTTING CHECKS, BONDS, &c.— John F. Cory and Jacob H. Brown, Brooklyn, E. D., N. Y.

Claim.—1. The spring punches A, provided with cutting -dies representing figures or characters, as set forth, combined with female counter-dies I arranged in a series, and in such a manner that the male and female dies are counterparts of each other respectively, substantially as described.

2. The punches A and bed-plate C, provided with female dies I, in combination with the sliding spring holder J, substantially as described.

From The Inland Merchant, volume 2 (1910) [combined snippets]:

An article which has so many ingenious combinations that it becomes a kind sublimated Robinson Crusoe's tool chest is the Kismit Multi-doo. It is a paper cutter that does a whole lot of things besides, and does them satisfactory. Here are some of the other purposes to which it can be put with good results : It will extract pens, cut checks, be used as a copy guide, note book rest, book opener, rule, envelope opener, rule, envelope opener, ink eraser, pen and pencil holder, pencil sharpener, book mark, and last but not least, an automatic paper clip. What more could a person ask of it? It is made of steel highly polished, and retails at 25 cents.

From Schnipper v. Township of North Bergen (New Jersey Superior Court, decided April 11, 1951), reprinted in Reports of Cases Argued and Determined in the Superior Court, Appellate Division, Chancery Division, Law Division, and in the County Courts of the State of New Jersey (1951):

Our review of the record satisfies us that Mrs. Schnipper did not sustain her burden in this case. Her duties consisted of typing the payroll, cutting payroll checks, sorting payroll vouchers, making entries in certain books and similar activities of a general clerical nature.

...

So far as appears, bookkeeping as generally understood is not part of the job; the only evidence concerning books is that appellant's duties included the making of some simple entries in cash disbursement books; her work and time were principally devoted to the typing of the payroll, the cutting of payroll checks and the sorting of payroll vouchers.

From what appears to be a caption in The Punched Card Machine Accounting and Data Processing Semi-annual (1952) [snippet view]:

Continuous forms payroll machine for signing, numbering, dating, trimming and cutting checks at high speed.

Also relevant is an ad for a hydraulic paper cutter designed to meet banking needs. The ad ran in both Burroughs Clearing House, volume 44 (1959), and Banking: Journal of the American Bankers Association, volume 52 (1960) [combined snippets]:

Especially designed for accurate, fast production such as cutting checks imprinted with Magnetic Characters, the Chandler & Price Full Hydraulic Power Paper Cutter with Automatic Spacing Attachment also handles cutting for all types of bank forms—ledger cards, statements, deposit slips, letterheads, etc. — speedily, easily and economically.

The Automatic Spacing Attachment provides the facility for repetitive cuts—a great time-saving factor in cutting imprinted checks—without requiring the operator to make precise, visual adjustments of the knife for each cut. Furthermore, absolute accuracy of each cut is assured—a very vital consideration in cutting checks printed with magnetic inks.

The constantly increasing volume of imprinted check production coupled with new methods of handling, sorting and recording requires fast, low-cost, accurate, cutting equipment. The Chandler & Price Full Hydraulic Power Paper Cutter with Automatic Spacing Attachment fully meets the needs of the bank maintaining a printing department.

Conclusions

Examples from Google Books indicate that "cut checks" goes back to the nineteenth century, both in the sense of "create or make out a check" and in the sense of "cancel or spindle a check." From the discussion in George Jackson's 1841 book on accounting, it appears that in that era checks were indeed cut from a master checkbook at need. Advertisements for automated check-cutting machinery have appeared as recently as 1960, suggesting that physical cutting of checks continued to occur (in some settings) more than a century after Jackson addressed the topic.

Even though some want to see a difference between normal form and progressive form I dare say that they see things that are not there. We invite you makes more sense than We are inviting. If I had to correct the above passage I would never say change we invite in we are inviting you. That only leads to the effect that a learner will ever be unsure whether to use the normal form or the progressive form.

To say that the progressive form is more appropriate is insufficient as there is no reason given. If the teacher had said you might as well use We are inviting you - not to emphasise the progressive aspect but to use it as a form that has a bit more weight - simply because it is longer - then I would agree with this teacher. But that is a choice you can decide on for yourself. But there is no rule that says in this sentence only I'm inviting you is correct.

You might as well consider another formula: We would be happy if you accepted our invitation for ...