Do EU residents need an A1 certificate for any short business trip in another EU country?

Note this answer has been heavily edited in the light of the OP's clarifications; some of the comments below will read better in the context of the earlier versions of this answer, which can be seen in the edit history.

I originally filleted the KPMG advice and the EC regulations to which it refers, but here's the rub: it doesn't actually matter whether your university has a statutory basis for the request. Your employers are entitled to ask you to do things over and above those required by law - that's why they pay you. If the university says you have to get a purple herring each time you go abroad, well, time to start printing purple herring requisition slips and sending them down to the office supplies department. Or the fishmonger.

As for the question

Do EU citizens need an A1 certificate for any business trip in another EU country?

I believe Her Majesty's Government imposes no such statutory requirement; certainly I've never been asked to acquire such a certificate in 20 years of travelling around the EU for business and pleasure. We have our EHIC cards, as I believe you do, but HMG's detailed page on these makes no mention of any need to track each trip to a fellow member state. I suppose that HMG has decided that the costs of tracking which of its citizens is actually abroad and enjoying theoretical foreign insurance cover on any given day vastly outweighs the costs of paying a few improper insurance claims, and they are probably right. It is certainly possible that that puts the UK in incomplete implementation of the regulations, but that's no defence you can use to change your employer's requirements - see above.

There are answers to this question here:

• eu - When do I need an A1 certificate? - Expatriates Stack Exchange

Since EU law stipulates that employees need be insured only once, this certificate is proof of this insurance.

• EU Regulation 987/2009

I understand that to make the local administration happy, I should fill a form to request this certificate for every EU travel 2 weeks before traveling to another EU country

• the goal is the prevention of employees being molested by demands from local insurance companies to prove that they are properly insured
• The employer will apply for the A1 Certificate when deploying their employee abroad from the employees insurer. The issued A1 Certificate will then be given to the employee to take with them.

Is the interpretation that an A1 certificate is necessary for any small business trip, really correct?

• It is needed for all employees from one EU country being temporarily deployed (up to 24 Months) to another, which includes small business trips

Maybe it is even necessary to distinguish between how this is handled "de jure", and "de facto".

• One cause is that it is relatively unknown, because it is relatively new system that envolved due to the difficulty of local health insurances to determine if the employee is exempted from being insured locally

Maybe it is also just some local German interpretation, and many are just coping this interpretation.

• 2018 the Health Insurance companies (Krankenkassen) in Germany , to avoid a flood of paper applications, started the automation of the issuing of these certificates. Starting 2019, the employers can electronically request them and paper applications can no longer be made since the 1st of July.
• How the Health Insurance companies in the other EU countries are implementing the 2009 directive, I cannot say

Summary:

The goal is that insurance inspector's can fullfil their duties in insuring that all employees are properly insured and to reduce the misuse of social services.

For an employee (who is already insured in their home country) being temporarily assigned to another country, there is no need to insure them again

• there are known cases where employees have been send abroad without any insurance at all
• or where the employer claims that they are, but in reality they are not.

The A1 Certificate is a verification from the insurer that the employee is properly insured by them for work abroad.

Since it is commonplace that the burdon of proof lays by the employee (as apposed to the insurance inspector), the A1 Certificate serves the purpose of supplying this proof.

As for the idea that insurance inspector's are being employed to lurk at refuelling stops or around meeting rooms to check peaples insurance coverage is extremely unrealistic (not to mention cost inefficient).