The link which you have shared allows the user to bypass the duplicate matching rule defined in salesforce. In REST API, the method will be same. Sharing a code snippet here for enforcing the insertion of duplicate records and fetching the already existing duplicate records.

The method simply creates an Account and returns the list of duplicate accounts existing in the database.

@HttpPost
global static List<Id> fetchDuplicates(String name, String phone, String website) {

    Set<Id> setDuplicateIds = new Set<id>();
    Account account = new Account(Name = name, phone = phone, website = website);
    Database.SaveResult sr = Database.insert(account, false);

    if (!sr.isSuccess()) {

        Datacloud.DuplicateResult duplicateResult;

        // Insertion failed due to duplicate detected
        for(Database.Error duplicateError : sr.getErrors()){

            duplicateResult = ((Database.DuplicateError)duplicateError).getDuplicateResult();
        }

    // Fetch the Ids of the existing duplicate records
        for(Datacloud.MatchResult duplicateMatchResult : duplicateResult.getMatchResults()) {

            for(Datacloud.MatchRecord duplicateMatchRecord : duplicateMatchResult.getMatchRecords()) {

                setDuplicateIds.add(duplicateMatchRecord.getRecord().Id);
            }
        }

        System.debug('Duplicate records ' + setDuplicateIds);

        // If the duplicate rule is an alert rule, we can try to bypass it
        Database.DMLOptions dml = new Database.DMLOptions(); 
        dml.DuplicateRuleHeader.AllowSave = true;
        Database.SaveResult sr2 = Database.insert(account, dml);

        if (sr2.isSuccess()) {
            System.debug('Duplicate account has been inserted in Salesforce!');
        }
    }

// Return the existing duplicate records Ids
    List<Id> lstDuplicateIds = new List<Id>();
    lstDuplicateIds.addAll(setDuplicateIds);
    return lstDuplicateIds;
}

Hope this helps!

I don't think you can add RecordType in the Matching Rule Criteria based on this feature request in Ideas:

Include Record Type as a field option for Matching Rules.

When setting up Matching Rules, there are times when you may want to allow duplicates values in a field if the record types don’t also match. To do this it would be helpful if we could also include the Record Type in the Matching Rule.