Learn English – “Do you have more classes” Vs. “Do you have another classes”

The simplest distinction is between "fabricating" and the others. "Fabricate" in the sense of "manufacture" has no moral or ethical implications, although it can also be used in the sense of making things up and, by implication, lying. A statement which is a fabrication (a lie) would not be called a counterfeit or forged.

"Forge" and "counterfeit" are usually distinguishable

1) In the case of currency, "counterfeit" is the preferred term.

2) In the case of art or signatures, "forge" is used. Here, the personal involvement of the faker is emphasized.

2a) As an exception, a forged work of art is sometimes referred to as counterfeit, such as "a counterfeit Rembrandt". Here, the use emphasizes the work as such, with the exact identity of the forger/counterfeiter being less important.

3) Objects which are not produced to fool the purchaser as to the personal source of the object are generally referred to as counterfeit. An example would be "counterfeit Gucci handbags", since although house Gucci controls the design, no one supposes that Guccio Gucci himself makes any of the products sold.

Note that 3) explains 1). Currency is not produced by individual artists or even Uncle Sam. As such, its impersonal nature of production establishes "counterfeit" as the preferred term for a fake.

Both sentences are correct but the underlying message is like you are asking them a confirmation from you if you are okay to start on that date, not a confirmation from the management. It will be redundant since you already have confirmed that you are willing to start on that date.

I think you should reconstruct your message to something like asking for a confirmation if the said date (June 10) is the final date of your first day so you can book your flight.