Your guess is correct.
From
http://www.howtostudykorean.com/unit1/unit-1-lessons-9-16/unit-1-lesson-9/
이다 is the plain form in the present tense:
나는 선생님이다 = I am a teacher
If the last letter of the noun before 이다 ends in a vowel, you can eliminate 이. For example:
나는 의사다 = I am a doctor
나는 의사이다 = I am a doctor
Both of the above can be seen as correct. Here, the pronunciation of “이” is merging with the pronunciation of the vowel in the noun. If you pronounce the two sentences above, you can see that there is very little difference.
Conversely, if the last letter of the noun before 이다 is a consonant, this merging cannot happen. For example:
나는 선생님이다 = I am a teacher (correct)
나는 선생님다 – incorrect
When conjugating 이다 into the past tense in the plain form, “었다” is added to the stem of “이다” (이). This is actually quite simple for you to understand, because every other verb and adjective follows this same rule. For example:
의사이었다
선생님이었다
However, the pronunciation of 이었다 can merge to “였다” when the noun that it is being attached to ends in a vowel. For example, both of these are correct:
의사이었다
의사였다
Pronounce both of those, and listen to how little of a difference there is between the two. Not only that, the pronunciation of both of those is very easy and it flows off the tongue.
Conversely, 이 and 었다 cannot merge when the noun it is added to ends in a consonant. For example:
선생님이었다 – correct
선생님였다 – incorrect
Pronounce both of those and listen the difference. Not only that,‘선생님였다’ is hard to pronounce. It is difficult to move your mouth from the ㅁ sound directly to the 여 sound. It is much easier to pronounce it like this: 나는 선생님-이-었-다.