Python requests - How to handle different exceptions correctly?

Question

Using requests module I query data from api's and I am trying to figure out how to handle exceptions correctly in order to count statistics on the errors that occured. I have the following piece of code to make the request:

import requests
from urllib3.exceptions import ReadTimeoutError

## class definition and some other code here...

def _make_get_request(self, api_url: str, payload=None) -> json:
    try:
        self._req_stats_dict.update({'api_req_get': self._req_stats_dict.get('api_req_get', 0) + 1})
        if payload:
            response = requests.get(api_url, params=payload, timeout=self._API_TIMEOUT)
        else:
            response = requests.get(api_url, timeout=self._API_TIMEOUT)
        response.raise_for_status()
        clogger.debug(response.json())
        if not response.json():
            self._req_stats_dict.update({'api_no_result': self._req_stats_dict.get('api_no_result', 0) + 1})
        # raise Exception("test api error")
        return response.json()
    except requests.exceptions.HTTPError as e:
        self._req_stats_dict.update({'api_no_response': self._req_stats_dict.get('api_no_response', 0) + 1})
        clogger.warning(f"<{self}>: {e} \n{payload}")
    except TimeoutError as e:
        clogger.warning('Api TimeoutError occured')
        self._req_stats_dict.update({'api_timeout': self._req_stats_dict.get('api_timeout', 0) + 1})
    except ReadTimeoutError as e:
        clogger.warning('Api ReadTimeoutError occured')
        self._req_stats_dict.update({'api_timeout': self._req_stats_dict.get('api_timeout', 0) + 1})
    except requests.exceptions.ReadTimeout as e:
        clogger.warning('Api requests.exceptions.ReadTimeout occured')
        self._req_stats_dict.update({'api_timeout': self._req_stats_dict.get('api_timeout', 0) + 1})
    except Exception as e:
        self._req_stats_dict.update({'api_error': self._req_stats_dict.get('api_error', 0) + 1})
        clogger.exception(f"{self} An unknown api error occured: {e} \n{payload}")

So when getting a timeout error, it looks like this:

WARNING | Api requests.exceptions.ReadTimeout occured
ERROR   | ApiSomehost: HTTPSConnectionPool(host='api.somehost.io', port=443): Read timed out. (read timeout=3)
Traceback (most recent call last):
  File "/opt/venv/lib/python3.10/site-packages/urllib3/connectionpool.py", line 449, in _make_request
    six.raise_from(e, None)
  File "<string>", line 3, in raise_from
  File "/opt/venv/lib/python3.10/site-packages/urllib3/connectionpool.py", line 444, in _make_request
    httplib_response = conn.getresponse()
  File "/usr/local/lib/python3.10/http/client.py", line 1374, in getresponse
    response.begin()
  File "/usr/local/lib/python3.10/http/client.py", line 318, in begin
    version, status, reason = self._read_status()
  File "/usr/local/lib/python3.10/http/client.py", line 279, in _read_status
    line = str(self.fp.readline(_MAXLINE + 1), "iso-8859-1")
  File "/usr/local/lib/python3.10/socket.py", line 705, in readinto
    return self._sock.recv_into(b)
  File "/usr/local/lib/python3.10/ssl.py", line 1273, in recv_into
    return self.read(nbytes, buffer)
  File "/usr/local/lib/python3.10/ssl.py", line 1129, in read
    return self._sslobj.read(len, buffer)
TimeoutError: The read operation timed out

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/opt/venv/lib/python3.10/site-packages/requests/adapters.py", line 439, in send
    resp = conn.urlopen(
  File "/opt/venv/lib/python3.10/site-packages/urllib3/connectionpool.py", line 785, in urlopen
    retries = retries.increment(
  File "/opt/venv/lib/python3.10/site-packages/urllib3/util/retry.py", line 550, in increment
    raise six.reraise(type(error), error, _stacktrace)
  File "/opt/venv/lib/python3.10/site-packages/urllib3/packages/six.py", line 770, in reraise
    raise value
  File "/opt/venv/lib/python3.10/site-packages/urllib3/connectionpool.py", line 703, in urlopen
    httplib_response = self._make_request(
  File "/opt/venv/lib/python3.10/site-packages/urllib3/connectionpool.py", line 451, in _make_request
    self._raise_timeout(err=e, url=url, timeout_value=read_timeout)
  File "/opt/venv/lib/python3.10/site-packages/urllib3/connectionpool.py", line 340, in _raise_timeout
    raise ReadTimeoutError(
urllib3.exceptions.ReadTimeoutError: HTTPSConnectionPool(host='api.somehost.io', port=443): Read timed out. (read timeout=3)

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/workspace/src/floor/floorapp/api/api_strategy.py", line 39, in _make_get_request
    response = requests.get(api_url, params=payload, timeout=self._API_TIMEOUT)
  File "/opt/venv/lib/python3.10/site-packages/requests/api.py", line 75, in get
    return request('get', url, params=params, **kwargs)
  File "/opt/venv/lib/python3.10/site-packages/requests/api.py", line 61, in request
    return session.request(method=method, url=url, **kwargs)
  File "/opt/venv/lib/python3.10/site-packages/requests/sessions.py", line 542, in request
    resp = self.send(prep, **send_kwargs)
  File "/opt/venv/lib/python3.10/site-packages/requests/sessions.py", line 655, in send
    r = adapter.send(request, **kwargs)
  File "/opt/venv/lib/python3.10/site-packages/requests/adapters.py", line 529, in send
    raise ReadTimeout(e, request=request)
requests.exceptions.ReadTimeout: HTTPSConnectionPool(host='api.somehost.io', port=443): Read timed out. (read timeout=3)

I don't quite understand this chain of tracebacks. It seems only the last exception requests.exceptions.ReadTimeout gets handled correctly, I am not sure what I am doing wrong with the other handlers? Would it be enough to handle just that exception and are the other handlers for TimeoutError and urllib3.exceptions.ReadTimeoutError necessary at all?

Thanks in advance!