How to understand time through sentences?

Solution 1:

When the verb "is" is used, that implies (absent indications to the contrary) the current time. It would not be improper to use "The current manager", as you suggest, but it would not normally be necessary, and it's wise to avoid unnecessary verbosity.

Solution 2:

Normally, if you don't specify which of the team's managers you're referring to, it would be understood as the current manager (in a sense, that is the default). The phrase "current manager" may come up more frequently in conversations where you may also be talking about other managers of the team through time and need to differentiate them, e.g., saying, "The current manager of team X is in Spain. The previous manager of team X is also in Spain. However, the next manager, who will be starting next Monday, is in Portugal at the moment."

Solution 3:

Hot Licks and auspicious99 are correct but don't give enough explanatory detail.

Is as mentioned refers to the period of time designated as now/current. Anything current can be identified with is (or am/are, depending on tense). John IS the manager, I AM the goalkeeper, Pete and Dave ARE the strikers.

Was designates time that occurred in the past. In your example, you would use was (or 'were' if plural) to show any of the many managers through time: Andrew WAS the manager in the 60's. The trio of Andy, Steve and Bob WERE the managers through the team's golden period.

For completeness will be is used for future references. As assistant manager, Michelle WILL BE the manager if John suddenly gets fired.