Syncing Salesforce data back to Call Tracking Metrics

Question

I'm currently using the salesforce integration in Call Tracking Metrics and triggers to sync up a leads/notes, etc.

But, for various reasons, I need to sync the notes (and possibly a few fields in the Lead) back...

I'm a junior dev, but relatively knew to Salesforce.

My only thought is to build a process automation that then sends an outbound message (only in SOAP for some reason? Not a REST?) to an API, parse the message and then hopefully have some per note ID that allows me to compare and write what is missing.

Is this "the way" or is there some automation I'm not aware of?

Further clarifications: How do I simply send a payload with select data to a REST API on specific triggers in Salesforce? There's a lot of names for things, but not necessarily transalted like "This is a REST API, This is a Database trigger"...

The equivalent in psuedo code would be:

var MySQLEvents = require('mysql-events');
var dsn = {
  host:     _dbhostname_,
  user:     _dbusername_,
  password: _dbpassword_,
};
var mysqlEventWatcher = MySQLEvents(dsn);
var watcher =mysqlEventWatcher.add(
  'myDB.table.field.value',
  function (oldRow, newRow, event) {
     //row updated
    if (oldRow !== null && newRow !== null) {
      fetch('https://exampleapi.com/apitodigest', {
      method: 'POST', // or 'PUT'
      headers: {
       'Content-Type': 'application/json',
      },
      body: JSON.stringify('DATA FROM SQL'),
    })
    .then((response) => response.json())
    .then((data) => {
      console.log('Success:', data);
    })
    .catch((error) => {
      console.error('Error:', error);
    });
    }
  }, 
);

Note, I'm not looking for code and hand holding (but I wouldn't mind), but I'm just which tool or marketing name represents the above.

Thank you...