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I'm using LMC6482 opamp with +/-15V rail voltages and having weird behavior. This means Vcc is +15V and Vee is -15V in my case.

Am I exceeding the specs? The jargon in the datasheet is not clear or should I say I'm not familiar with it.

Yes, you are exceeding the specs. Look at table 6.1 of the DATASHEET on page 3:

It clearly states the maximum difference between the V+ and V- pins is 16V. This means you can have +8V and -8V, as the difference is 16V.

As you have it right now, your difference is 30V, which is almost double the maximum ratings. Chances are, you have damaged the op-amp and it will need to be replaced. You should ALWAYS check the maximum ratings of any IC you are using if you are worried. If you can, try and be within the maximum, so +7.5V and -7.5V would work nicely here, for a total of 15V supply.

Figure 1. V_max.

This line of the datasheet is stating that the maximum voltage difference between V⁺ and V^- is 16 V.

You have probably damaged the op-amp.

I read your question as "yes I've read the datasheet, but its unclear". So let's address how to read the supply voltage range specification as the manufacturer Texas Instruments write it in their original product data sheet:

Supply Voltage (V⁺ - V^-)... MAX 16V

You have a V⁺ of 15V and a V^- of -15V.
Now we use the formula from the data sheet and get:

(15 - (-15)) = 15 + 15 = 30V

That's the way to interpret this voltage range specification. So yes, you are exceeding specified maximum supply voltage range by 14V.

Other answers mentione the asbsolute maximum ratings, which is NOT what youy should look for when designing a functional system. Instead, you should use the recommended operating conditions. In this case, those specify 15.5V.

In your case, you could check the absolute maxima to find out whether your chip had a chance of survival. As others have pointed out, that is not the case.