I think "the" would always be included before such words as "Customer" or "Company".
No, it's not always included before such words. Regardless of random examples, there are other examples where "the" is not included. I used to feel as you do and always included the article. Then I got a harsh complaint from a customer who said they had never before seen a contract written that way. I had to "correct" my translation and remove the article preceding all the defined parties. Since then I have been hesitant, because I'm not sure if there is a standard rule on this.
One can make the argument that if we would not say "the Mr. Smith", then we should not say "the Customer" when we are using Customer to replace Mr. Smith. If we're referring to an unnamed customer, then it would be the customer, but if we're referring to Mr. Smith, then it's "Customer". Just as we would say "Article 10", but "the article referred to above".
Here are a few guides that say the opposite of what Kenneth A. Adams says above:
Defining Individual Parties
Make sure the term is defined in the beginning of the contract, in a clause added for introducing the parties. Do not use articles for this term, such as “an” or “the.” In other words, don't say “the seller,” just say “Seller.” This is done because you are simply replacing the party's name with the defined term.
The parties defined
To refer to a contract party in the agreement, either use the functional reference (e.g. Seller, Licencee, Service Provider, Lender) or the short name of the party (e.g. Weagree, Shell, Philips, Sony).
Going by that rule, omitting the article has nothing to do with search-and-replace problems, as per Mr. Adams, but rather with the fact that the term is replacing the name of the party, and as such becomes a proper name in itself. I'm not saying I think it's the correct usage. I'm merely saying it's possible to argue for the rule.
The American Bar Association agrees with Adams that using the article makes a contract marginally easier to read. But they appear to be saying they prefer this, but it's not a rule:
the front of the contract - American Bar Association
See Article 1.73:
If a party name consists of a common noun, using the definite article—the Purchaser rather than Purchaser—results in prose that's marginally less stilted. In any event, be consistent throughout a contract in using or not using the definite article.
This is why I'm asking the question. Is there a standard usage, or not? Valid arguments can be made for both. From our combined research, it looks to me like it's chiefly a matter of personal preference. But perhaps the question is about context rather than standard usage. Could it be that in some countries using the article is more prevalent, or that it depends on the type of contract?
I've been continuing my research and came across a fascinating discussion on the blog of the very same Ken Adams:
“The Vendor” or “Vendor” — A Practical Consideration
Also, the same Kenneth A. Adams, on page 9 of his book
Legal Usage in Drafting Corporate Agreements, Greenwood Publishing Group, 2001, states the following
Defined Terms in the Introductory Clause
If you elect to use a common noun as the defined term for a party name, you must decide whether to use a definite article. Once you have made your choice, be consistent; a surefire way to look like a sloppy drafter is to alternate randomly between, for example, Purchaser and the Purchaser. Doing without the definite article results in prose that is marginally more stilted than it would be with the definite article, but it also results in a marginally shorter document. I generally prefer using the definite article.